Sir Issac Newton Notes
Introduction to the Language of Kinematics
Describing Motion with Words
Scalars and Vectors
Distance and Displacement
Speed
Velocity
Acceleration
Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations. The goal of any study of kinematics is to develop sophisticated models which serve in describing (and ultimately, explaining) the motion of real-world objects.
Much of our lives are spent in motion, travelling from one place to another. We measure motion by the rate of change that occurs in a certain amount of time.
Scalars and Vectors
Physics is a mathematical science - that is, the underlying concepts and principles have a mathematical basis. The motion of objects can be described by words - words such as distance, displacement, speed, velocity, and acceleration. These mathematical quantities that are used to describe the motion of objects can be divided into two categories. The quantity is either a vector or a scalar. These two categories can be distinguished from one another by their distinct definitions:
Scalars are quantities with only magnitude
Vectors are quantities having both a magnitude and a direction.
Distance and Displacement
The definition of displacement is the change in an object's position. Describing the change in an object's position entails two things: 1) How far did the object go? 2) And in what direction? The direction part is easy. In one-dimensional motion there are only two directions; the positive direction and the negative direction. If the object ends up in the positive direction from where it started it, its displacement is positive. If it ends up in the negative direction from where it started, its displacement is negative.
The only thing that matters when you are talking about displacement is where the object started its motion and where it ended it, NOT where it went in between.
Distance is a scalar measure of the interval between two locations measured along the actual path connecting them. Displacement is a vector measure of the interval between two locations measured long the shortest path connecting them. For example, as illustrated in figure 1, a man rides his bicycle from Manhattan to New Jersey traveling over the Hudson River.
Getting there is a three step process.
1. Follow the Hudson River 8.2 km upriver.
2. Cross using the George Washington
Bridge bike path - 1.8 km between anchorages. 3 Reverse direction and head downriver for 4.5 km.
The distance traveled is 14.5 km, but the resulting displacement is a mere 3.7 km north. The end of this journey is actually visible from the start. Maybe he should buy a canoe.
There is an easy way to mathematically calculate the displacement of an object during any motion. Get the position (on the axis) of the object at the beginning of its motion and the position (on the same axis) of the object at the end of its motion. Be sure to include the positive or negative sign with the position. Then subtract the starting position (the initial position, represented as xi) from the ending position (the final position, represented as xf). This gives you the displacement, including its direction (positive or negative, based on the sign of the answer). Δx =xf-xi
Speed
Speed is a scalar quantity that refers to "how fast an object is moving. Speed is the distance an object travels in a certain amount of time. The speed of an object is constant when the object travels in equal distances in equal periods of time. We define the average speed as the total distance, s, traveled during a particular time divided by the time interval, t. To calculate speed you simply need to divide the distance traveled by the time it took to complete the trip. If the average speed is the same for all parts of travel, then the speed is constant. However, since a moving object often changes its speed during its motion, it is common to distinguish between the average speed and the instantaneous speed. The distinction is as follows:
Instantaneous Speed - speed at any given instant in time
Average Speed - average of all instantaneous speeds, found simply by a distance/time ratio.
Velocity
Velocity is a vector quantity that refers to "the rate at which an object changes its position." It is the displacement or position change during a particular time interval. The task of describing the direction of the velocity vector is easy! The direction of the velocity vector is simply the direction that an object is moving. Average velocity is computed using the equation:
When determining the velocity of an object, the direction must be known. It would not be sufficient to say an object’s velocity is 55 miles/hr. The object's velocity would need to include a direction, 55 miles/hr, east.
What is the difference between
Speed and Velocity?
To specify a velocity, we must give both its size or magnitude (how fast) and its direction (north, south, east, west up, down, right, and left) If you say that an object is moving 15m/s, you have stated its speed. If you say that the object is moving due west at 15m/s, you have indicated its velocity.
As can be seen in Figure 2, the changes in the velocity of the car were produced by forces acting upon the car. The most important force involved in changing the velocity of the car is the frictional force exerted on the tires of the cart by the road surface. A force is required to change either the size or the direction of the velocity. If no net force were acting in the car, it would continue to move at a constant speed in a straight line.
What is a Vector
Has stated previously, a vector is a quantity for which both size and direction are important. Many of the quantities used in describing motion are vector quantities such as velocity, acceleration, force and momentum to name a few.
The length of the arrow is drawn proportional to its size. In other words, the larger the velocity, the longer the arrow will be (fig. 4).
Figure 4 The length of the arrow shows the size of the velocity vector
Acceleration
Acceleration is a familiar idea. We use the term in speaking of the acceleration of a car driving away from a traffic light, or the acceleration of a running back in football. We feel the effects of acceleration on our bodies when a car’s velocity changes rapidly and even mores strikingly when an elevator lurches upward, leaving our stomachs slightly behind. These are all forms of accelerations.
Understanding acceleration is crucial to the study of motion. Acceleration is a vector quantity that is defined as:
“the rate at which an object changes its velocity."
Where a is the acceleration, v1 is the starting velocity, v2 is the ending velocity, and t is the time
An object is accelerating if it is changing its velocity. For example, if a car is headed due north at 10m/sec and you step on the accelerator and increase its velocity to 20 m/sec in 5 sec, you can compute the rate of acceleration as follows:
The final result is read as 2 meters per second per second or 2 meters per second squared. (2m/s2). Assuming your acceleration is uniform, you may interpret the calculation as, the car started out at 10m/sec, but one second later the car increased to 12m/sec; 2 seconds later, 14m/sec; 3 seconds later, 16m/sec; 4 seconds later, 18m/sec; and 5 seconds later, 20m/sec. Therefore, the car increased its velocity 2 meters per second, every second.
The data in the table above is representative of an object accelerating - the velocity is changing with respect to time. In fact, the velocity is changing by a constant amount, 10 m/s in each second of time. Anytime an object's velocity is changing, that object is said to be accelerating
Sometimes an accelerating object will change its velocity by the same amount each second. As mentioned in the above paragraph, the data above shows an object changing its velocity by 10 m/s in each consecutive second. This is referred to as a constant acceleration since the velocity is changing by a constant amount each second. An object with a constant acceleration should not be confused with an object with a constant velocity. Don't be fooled! If an object is changing its velocity - whether by a steady or a varying amount - then it is an accelerating object. In addition, an object with a constant velocity is not accelerating. The data tables below depict motions of objects with a constant acceleration and a changing acceleration. Note that each object has a changing velocity.
Describing Motion with Graphs
The Meaning of Shape for Velocity vs. Time Graph
The velocity vs. time graphs for the two types of motion - constant velocity and changing velocity (acceleration) - can be summarized as follows.
Positive Velocity
Positive Velocity
Zero Acceleration
Positive Acceleration
The shapes of the velocity vs. time graphs for these two basic types of motion - constant velocity motion and accelerated motion (i.e., changing velocity) - reveal an important principle. The principle is that the slope of the line on a velocity-time graph reveals useful information about the acceleration of the object. If the acceleration is zero, then the slope is zero (i.e., a horizontal line). If the acceleration is positive, then the slope is positive (i.e., an upward sloping line). If the acceleration is negative, then the slope is negative (i.e., a downward sloping line). This very principle can be extended to any motion conceivable. The slope of a velocity-time graph reveals information about the object's acceleration. But how can one tell whether the object is moving in the positive direction (positive velocity) or in the negative direction (negative velocity)? And how can one tell if the object is speeding up or slowing down?
Since the graph is a velocity-time graph, the velocity would be positive whenever the line lies in the positive region (above the x-axis) of the graph. Similarly, the velocity would be negative whenever the line lies in the negative region (below the x-axis) of the graph. Finally, if the line crosses over the x-axis from the positive region to the negative region of the graph (or vice versa) then the object has changed its directions.
Now how can one tell if the object is speeding up or slowing down? Speeding up means that the magnitude (the value) of the velocity is getting larger. For instance, an object with a velocity changing from +3 m/s to + 9 m/s is speeding up. Similarly, an object with a velocity changing from -3 m/s to –9 m/s is also speeding up. In each case, the magnitude of the velocity (the number itself, not the sign) is increasing; the speed is getting bigger. Given this fact, one would believe that an object is speeding up if the line on a velocity-time graph is changing from near the 0-velocity point to a location further away from the 0-velocity point. If the line is moving away from the x-axis (the 0-velocity point) then the object is speeding up, and if the line is moving towards the x-axis, then the object is slowing down.
Accelerating object are changing their velocity
The general RULE OF THUMB is:
If an object is slowing down, then its acceleration is in the opposite direction of its motion. This RULE OF THUMB can be applied to determine whether the sign of acceleration of an object is positive or negative, right or left, up or down, etc. Consider the two data tables below. In each case, the acceleration of the object is in the positive "+" direction. In Example A, the object is moving in the positive direction (i.e., has a positive velocity) and is speeding up. When an object is speeding up, the acceleration is in the same direction as the velocity. Thus, this object has a positive acceleration. In Example B, the object is moving in the negative direction (i.e., has a negative velocity) and is slowing down. According to our RULE OF THUMB, when an object is slowing down, the acceleration is in the opposite direction as the velocity. Thus, this object also has a positive acceleration.
Example A
Positive direction speeding up
Example B
Negative direction slowing down
Time
(s)
Velocity
(m/s)
Time
(s)
Velocity (m/s)
0
0
0
-8
1
2
1
-6
2
4
2
-4
3
6
3
-2
4
8
4
0
Both are examples of positive acceleration
This same RULE OF THUMB can be applied to the motion of the objects represented in the two data tables below. In each case, the acceleration of the object is in the negative (-) direction. In Example C, the object is moving in the positive direction (i.e., has a positive velocity) and is slowing down. According to our RULE OF THUMB, when an object is slowing down, the acceleration is in the apposite direction as the velocity. Thus, this object has a negative acceleration. In Example D, the object is moving in the negative direction (i.e., has a negative velocity) and is speeding up. When an object is speeding up, the acceleration is in the same direction as the velocity. Thus, this object also has a negative acceleration. Example C
Positive direction slowing down
Example D
Negative direction speeding up
Time
(s)
Velocity
(m/s)
Time
(s)
Velocity (m/s)
0
8
0
0
1
6
1
-2
2
4
2
-4
3
2
3
-6
4
0
4
-8
Both are examples of negative acceleration
The slope of a velocity-time graph reveals information about the object's acceleration. But how can one tell whether the object is moving in the positive direction (positive velocity) or in the negative direction (negative velocity)? And how can one tell if the object is speeding up or slowing down?
Since the graph is a velocity-time graph, the velocity would be positive whenever the line lies in the positive region (above the x-axis) of the graph. Similarly, the velocity would be negative whenever the line lies in the negative region (below the x-axis) of the graph. And finally, if a line crosses over the x-axis from the positive region to the negative region of the graph (or vice versa), then the object has changed directions.
Now how can one tell if the object is speeding up or slowing down? Speeding up means that the magnitude (the value) of the velocity is getting larger. For instance, an object with a velocity changing from +3 m/s to + 9 m/s is speeding up. Similarly, an object with a velocity changing from -3 m/s to –9 m/s is also speeding up. In each case, the magnitude of the velocity (the number itself, not the sign) is increasing; the speed is getting bigger. Given this fact, one would believe that an object is speeding up if the line on a velocity-time graph is changing from near the 0-velocity point to a location further away from the 0-velocity point. That is, if the line is moving away from the x-axis (the 0-velocity point), then the object is speeding up. Conversely, if the line is moving towards the x-axis, then the object is slowing down.
Force and Acceleration
Is there a relationship between force and acceleration? An applied force is what causes an object to accelerate. The force along with the mass of an object determines the degree of response (acceleration) that the object experiences as a result of the force, but remember acceleration does not cause a force.
Mass
In the formula, a = F/m, it is also easier to determine the effect of mass on the amount of acceleration that an object experiences as a result of an applied force. Since mass (m) appears on the bottom, we know that mass is inversely related to acceleration. What this means is that if the same amount of force is applied to two objects, the object with the larger mass will experience a smaller amount of acceleration (or response to the force).
Forces Are Additive
For this discussion, consider motion in one dimension. If you and a friend push a box with the same amount of force but in opposite directions, the box will not move, the forces cancel one another. As an example to further illustrate the point, you already know that it is easier to push a stalled car when two people are pushing it instead of just one person. Well, this is obvious you might say, but it provides an example of the additive nature of forces. Let us assume that it takes 50 Newtons of force to move a stalled car. If you were just pushing it by yourself, you would have to provide all of this force by yourself. However, if two people were pushing the car, your combined total would have to be 50 Newtons. Therefore, you could push 20 Newtons, and your friend could push 30 Newtons so that the total force would still be 50 Newtons.
Direction
We know that acceleration involves a direction because the definition of acceleration is a change in velocity or a change in direction over time. Likewise, force has a direction, namely the direction in which it is applied. Therefore, the direction in which the force is applied is also the direction in which the object accelerates. This should be obvious because the acceleration that an object experiences is the direct response to the applied force. Therefore, a force causes an object to change its velocity. Objects in nature tend to maintain a constant velocity, unless they have contact with a nonzero net force.
Acceleration in Detail
If you are still having difficulty determining the direction of the acceleration, it may be helpful to figure out the direction of the force first. Once the direction of the force is known, the direction of the acceleration will be in the same direction. For example, consider an object that is moving to the right and speeding up. Is the object accelerating? If so, in what direction is the object's acceleration?
Well, the first question is easily answered. All we have to do is recall the definition of acceleration. Anytime an object's velocity is changing; the object is also accelerating. Since velocity involves both direction and speed, if either of these is changing, then the velocity is changing. In this case, the speed is changing (increasing) while the direction is remaining constant; therefore the object's velocity is changing so the object is accelerating.
Now we know the object is accelerating, but in which direction is the force? If the object is moving to the right and speeding up then the object's velocity is changing (object is accelerating), so there must be a force. This is because of Newton's 1st Law of Motion; if an object is not moving at a constant velocity, then there must be a net force acting on that object. Now that we have established there is a net force acting on the object, then determine the direction that the force is exerted. Picture a car in neutral moving to the right. Which direction would you have to push the car to get it to speed up while moving to the right? The answer is that you would have to push to the right. Since we now know that the direction of the force is to the right and that the direction of the force and acceleration must be in the same direction, then the direction of the acceleration is to the right. This example involves an object that is "accelerating" instead of "decelerating".
The first conclusion that can be drawn is that, when the force is in the same direction as the velocity, the object will speed up in the same direction as the velocity. In other words, when the direction of the force is in the same direction as the velocity, the object "accelerates". In this example, the direction of the velocity is to the right because the object was moving to the right. This is in the same direction as the force, which was pointing to the right, as well.
The second conclusion that one can draw from this example is that an object speeds up in the direction of its velocity when the direction of its acceleration and its velocity are the same. Alternatively, one could say that, when the direction of acceleration is the same as the direction of an object's velocity, the object will "accelerate". This obviously follows from the first conclusion because we know that the direction of the force and acceleration are the same.
Example 2
Consider an object moving to the right and slowing down. Is the object accelerating? If so, in which direction is the object's acceleration?
Once again, the first question is easily answered. Since the velocity is changing in this example, the object is accelerating. Because the speed is changing (decreasing), the velocity is changing; therefore the object must be accelerating.
To answer the second question, it is, once again, easier to think of the direction of the force. Rephrase the questions above, if an object is moving to the right and slowing down, is there a net force present? If so, in which direction does this net force have to be in order to slow down the object while it is moving to the right?
Well, since the object's velocity is not constant, there must be a net force acting on that object. Consider a car moving to the right in neutral. In which direction would the force have to act in order for it to slow down? The answer is that the force would have to be to the left, or in the opposite direction. If we know the direction of the force, then we know the direction of the acceleration, as well, because they are always in the same direction. Therefore, the direction of the acceleration points to the left.
In summary, when the net force and the velocity are in opposite directions, as is true in this example, the object slows down or "decelerates". Likewise, we can say that an object "decelerates" when the direction of its acceleration is opposite to that of its velocity. Therefore, when we say an object is "decelerating", what we really are saying is that the direction of its acceleration is opposite to the direction of its velocity.
If the force remains in the same direction after the object's velocity drops to zero, the object will begin to speed up to the left. After the velocity drops to zero, the force and the velocity are in the same direction.
Therefore, if a net force is applied in a direction opposite to the object's velocity, the object will initially continue moving in the same direction while slowing down to zero. Then the object will begin to speed up in the direction of the force, which is opposite to the direction in which it was initially moving. In other words, when a net force is applied opposite to the velocity of an object, the object will initially "decelerate" to zero (while moving in the same direction that it was initially moving) and then it will begin to "accelerate" in the direction opposite to the initial direction that the object was moving in.
In summary:
1. When the net force is in the same direction as the object's velocity, the object will continue to move in the same direction, while its speed will increase. In other words, the object will "accelerate".
2. When the net force is opposite to the direction of the object's velocity, the object will initially continue to move in the same direction while slowing down. After the object reaches zero velocity, it will begin to speed up in the direction opposite to the direction of its initial velocity. In other words, the object will initially "decelerate" while moving in the same direction. Once the object "decelerates" to zero, (stops) it will then "accelerate" in the direction opposite to its initial velocity.
3. You might have wondered why I didn't consider the case of zero net force. In fact, we have already considered this possibility. This possibility is exactly what Newton's 1st Law of Motion describes. If an object is subjected to zero net force, then it will continue to move at a constant velocity, meaning that it will continue to move in a straight line at a constant speed. Furthermore, it will continue to move until it is subject to a nonzero net force.
Free Fall and the Acceleration of Gravity
A free-falling object is an object that is falling under the sole influence of gravity. It is a remarkable fact, first discovered over 300 years ago by Galileo that objects in free fall motion descend at a constant acceleration. During each second of the fall, the object gains 9.8 m/s2 (32 ft/s2) in velocity. This means that under free fall all objects have the same constant, acceleration equal to 9.8 m/s2. The acceleration of a free-falling object is known as acceleration due to gravity, and is denoted by the symbol g.
If you dropped, an object is the falling object accelerating. Think for a moment about whether the velocity is changing. Before you release an object, its velocity is zero, but the instant after the object is released, the velocity has some value different from zero. There has been a change in velocity. If the velocity is changing, there is acceleration.
Things happen so rapidly that it is difficult, just from watching the fall, to say much about the acceleration. It does appear to be large, because the velocity increases rapidly. Does the object reach a large velocity instantly, or does the acceleration occur more uniformly?
To answer this question, we must slow the motion down somehow so that our eyes and brains can keep up with what is happening. There are several ways to slow down the action. One is by using a stroboscope, a rapidly blinking light whose flashes occur at regular intervals in time. In Figure 5, a photograph taken using a stroboscope is used to illuminate an object as it falls. The position of the object is pinpointed every time the light flashes. If you look closely you will notice the distance covered is successive time intervals increases regularly. The time intervals between successive positions of the ball are all equal.
Figure 5
Time
Distance
Velocity
Time
Distance
Velocity
1
0
1.2cm
24cm/s
5
0.20s
19.7cm
218cm/s
2
0.05s
4.8cm
72cm/s
6
0.25s
30.6cm
268cm/s
3
0.10s
11.0cm
124cm/s
7
0.30s
44.0cm
320cm/s
4
0.15s
19.7cm
174cm/s
8
0.35s
60.0cm
368cm/s
You could verify the other values shown in the third column of table 3.1 by doing similar computations. It is clear in table above that the velocity values steadily increase. To see that velocity is increasing at a constant rate, we can plot velocity against time (fig. 3.4). Notice that each velocity data point is plotted at the midpoint between the two times (or flashes) from which it was computed. This is because these values represent the average velocity for the short time intervals between flashes
Notice that the slope of the line is constant in the graph above. The velocity values all fall approximately on a constant-slope straight line. Since acceleration is the slope of the velocity-versus-time graph, the acceleration must also be constant. The velocity increases uniformly with time.
To find the value of the acceleration, we choose two velocity values that lie on the straight line and calculate how rapidly the velocity is changing. For example, the last velocity value, 368 cm/s, and the second value, 72 cm/s, are separated by a time interval corresponding to 6 flashes or 0.30 second. The increase in velocity Δv is found by subtracting 72 cm/s from 368 cm/s, obtaining 296 cm/s. To find the acceleration, we divide this change in velocity by the time interval
cm/s2 = 9.8 m/s2
This result gives us the acceleration due to gravity for objects falling near the earth’s surface. Its value actually varies slightly from point to point on the earth’s surface because of differences in altitude and other effects. This acceleration is used so often that it is given its own symbol
g where g =9.8 m/s2
Called the gravitational acceleration or acceleration due to gravity, it is valid only near the surface of the earth and thus is not a fundamental constant.
The diagram below shows the position of a free-falling object every second. The fact that the distance, which the object travels every second, is increasing is a sure sign that the object is speeding up as it falls downward. Remember that if an object travels downward and speeds up, then its acceleration is downward. g = 9.8 m/s/s, downward
Figure 6
Figure 7 Free fall acceleration: Graphs of how the position (a), velocity (b), and acceleration (c) of objects under free fall motion change in time.
Acceleration Due to Gravity
Galileo predicted that heavy objects and light ones would fall at the same rate. The reason for this is simple. Suppose the coin has 50 times as much mass as the feather. This means that the earth pulls 50 times as hard on the coin as it does on the feather. You might think this would cause the coin to fall faster. But because of the coin's greater mass, it's also much harder to accelerate the coin than the feather - 50 times harder, in fact! The two effects exactly cancel out, and the two objects therefore fall with the same acceleration.
Near the surface of the earth, gravity is a downward acceleration. Over time, it causes all [unsupported] objects to pick up a downward speed.
Since gravity is a persistent acceleration, falling objects keep falling faster and faster.
Gravity accelerates objects towards the earth at 9.8 meters per second per second
This means each second it falls, it will be traveling 9.8 meters per second faster
If something is dropped from 227.00 meters (744.75 feet or 0.14 miles), it will hit the ground in 6.81 seconds
It will be traveling at 66.70 meters (149.21 mph) per second when it hits the ground
Falling with Air Resistance
This rule holds true only if gravity is the only force acting on the two objects. As an object falls through air, it usually encounters some degree of air resistance. Air resistance is the result of collisions of the object's leading surface with air molecules. The actual amount of air resistance encountered by the object is dependent upon a variety of factors: the speed of the object and the cross-sectional area of the object.
Increased speeds result in an increased amount of air resistance. Increased cross-sectional areas result in an increased amount of air resistance.
As an object falls, it picks up speed. The increase in speed leads to an increase in the amount of air resistance. Eventually, the force of air resistance becomes large enough to balances the force of gravity. At this instant in time, the net force is 0 Newtons; the object will stop accelerating. The object is said to have "reached a terminal velocity." As stated above, the amount of air resistance depends upon the speed of the object. A falling object will continue to accelerate to higher speeds until they encounter an amount of air resistance that is equal to their weight. Since the 150-kg skydiver weighs more (experiences a greater force of gravity), he will accelerate to higher speeds before reaching a terminal velocity. Thus, more massive objects fall faster than less massive objects because they are acted upon by a larger force of gravity; for this reason, they accelerate to higher speeds until the air resistance force equals the gravity force.
Representing Free Fall by Graphs
Free-falling motion can be represented using two basic graphs; position vs. time and velocity vs. time graphs.
A position vs. time graph for a free-falling object is shown below.
Observe that the line on the graph curves. As learned earlier, a curved line on a position vs. time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration g =10 m/s/s = approximate value), it would be expected that its position-time graph would be curved. A further look at the position-time graph reveals that the object starts with a small velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs. time graph is the velocity of the object the small initial slope indicates a small initial velocity and the large final slope indicates a large final velocity. Finally, the negative slope of the line indicates a negative (i.e., downward) velocity.
A velocity vs. time graph for a free-falling object is shown below.
Observe that the line on the graph is a straight, diagonal line. As learned earlier, a diagonal line on a velocity vs. time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 10 m/s/s), it would be expected that its velocity-time graph would be diagonal. A further look at the velocity-time graph reveals that the object starts with a zero velocity (as read from the graph) and finishes with a large, negative velocity; that is, the object is moving in the negative direction and speeding up. An object which is moving in the negative direction and speeding up is said to have a negative acceleration (if necessary, review the vector nature of acceleration). Since the slope of any velocity vs. time graph is the acceleration of the object, the constant, negative slope indicates a constant, negative acceleration. This analysis of the slope on the graph is consistent with the motion of a free-falling object - an object moving with a constant acceleration of 10 m/s/s in the downward direction.
Newton's Laws of Motion
First Law of Motion An object that is at rest will remain at rest unless a nonzero net (or total) force is exerted on it.
Newton's First Law, is also called the Law of inertia. Inertia is a tendency of an object to resist change in its state of motion. More massive objects have more inertia; that is, they have more tendencies to resist changes in the way they are moving. An elephant has a lot of inertia, for example. If it is at rest, it offers a large resistance to changes in its state of rest, and so it is difficult to move an elephant. On the other hand, a pencil has a small amount of inertia. It is easy to move a pencil from its state of rest. More massive objects have more inertia and thus require more force in order to change their state of motion. This one is easy to believe and sounds intuitive enough. However, the next part of the first law might sound less plausible. Simply stated, an object moving at a constant velocity will continue to move at a constant velocity (moving at a constant speed and in a straight line) unless a nonzero net (or total) force acts upon the object. Recall that constant velocity means that the object is moving at a constant speed and in a constant direction. You might think about something you’ve seen in the world that contradicts this statement. For instance, think of a car rolling in a straight line while in neutral. If the statement above is true, then the car should be able to roll in a straight line at a constant speed forever. This is obviously not true in real life because we know the car will eventually come to a stop. This certainly seems to contradict Newton's first law, or does it? Newton's first law states that an object moving at a constant velocity will continue moving in that fashion unless a nonzero net force acts upon that object. It is true that the car would continue to move in a straight line at a constant speed if there was no net force acting on the car. However, is there really no net force acting on the car? The force responsible for slowing down the car is friction. Therefore, the above observation, a car slowing down while in neutral, is not inconsistent with Newton's first law of motion.
In summary, objects like to move in straight lines, at constant speeds unless a force acts upon them. In fact, when a force acts on an object, the force causes the object to change its velocity. In other words, forces cause objects to accelerate.
Second Law of Motion
The acceleration of an object is directly proportional to the magnitude of the imposed force and inversely proportional to the mass of the object. The acceleration is in the same direction as that of the imposed force.
Simply stated, a force causes an object to accelerate. Whenever you see an object accelerating, there must be an external force acting on the object because, as stated in Newton's first law, objects move at a constant velocity unless acted upon by an outside force.
Mathematically, Newton's second law of motion can be expressed by the following formula: A = F/M where
A = acceleration,
F = force, and
M = mass.
What this formula tells us is that force causes an object to accelerate. However, it also tells us that the acceleration an object feels, in response to an applied force, does not solely depend on the amount of force applied. It also depends on the mass or inertia of that object. It also tells us that the more mass an object has, the less it accelerates in response to an applied force. This makes intuitive sense. For instance, if I apply the same force to a cotton ball and a book, the cotton ball would experience a greater acceleration than the book because the book has much more mass or inertia. Therefore, an object with a greater mass has a better tendency to resist a change in its motion when an external force is applied to that object. In other words, we say that the book has more inertia than the cotton ball.
Third Law of Motion
Newton's third law states that whenever a force is exerted, an equal and opposite force arises in reaction to this force. In other words, every force has an equal and opposite reaction force.
For example, when you push on a wall, the wall will also push back on you with an equal and opposite force. If this is true, why don’t you move backwards if the wall is pushing you backwards? The reason you do not move backwards when you push against a wall is that static friction is pushing you back with an equal amount of force to the right so that you do not move anywhere. In the above example, static friction would be pushing the person to the right with five Newton’s (5 N) of force, so that the person would experience zero net force, hence the person does not move. Therefore, if Newton's third law is true, and the wall pushes back on us just as hard as we push back on it,
Amusement Park Physics
When you enter an amusement park, you are walking into the world of physics. Some of us have found the nerve to ride a roller coaster or perhaps drop from a 35-story tower in free fall. Amusement park rides are designed to provide the rider with spine-chilling experiences. The laws of physics govern the extent of the thrill we experience. If we examine the physics involved in a roller coaster ride, the following properties can be observed:
Free-fall rides are made up of three distinct parts. During the first part of the ride, force is applied to the car to lift it to the top of the free-fall tower. As the rider climbs up a steep incline, they feel pressed against their seats; this provides a sense of weight. However, at the top of the tower, a 100-pound rider would feel 100 pounds of force from the seat pushing an external force upon their body. The rider feels their normal weight. The amount of force that must be applied depends on the mass of the car and its passengers. Motors apply the force, and there is a built-in safety allowance for variations in the mass of the riders. From the moment you begin climbing up the incline, you are building up potential energy, which is the stored energy that has the potential to be converted into kinetic energy. Potential energy is the energy possessed by an object because of its height above the ground. The amount of potential energy possessed by an object depends on its mass and its height. A roller coaster car is initially hauled by a motor and chain system to the top of a tall hill, giving it a large quantity of potential energy.
Suddenly the cart drops, plunging down the incline, accelerating toward the ground under the sole influence of gravity. As the rider free falls from the tower, they fall at the same rate as their surroundings-in this case, their seat. It is as though the seat has fallen out from under them; the rider no longer feels the external force of the seat and subsequently has a brief sensation of weightlessness. They have not lost any weight, but feels as though they have because of the absence of the seat force. In this context, weightlessness is a sensation and not an actual change in weight.
As the cart free falls, it is accelerating and is measured in g’s. A g is a unit of acceleration equal to the acceleration caused by gravity. Gravity causes free-falling objects on the earth to change their speeds at rates of about 10 m/s each second. That would be equivalent to a change in speed of 32 feet per second in each consecutive second. If an object is said to experience 3 g's of acceleration, then the object is changing its speed at a rate of about 30 m/s every second.
Once you begin your descent downwards, that energy is being converted into kinetic energy which is the energy of motion. So all of that potential energy that was stored up became kinetic energy. Kinetic energy is the energy possessed by an object because of its motion. All moving objects have kinetic energy. The amount of kinetic energy depends upon the mass and speed of the object. A roller coaster car has a lot of kinetic energy if it is moving fast and has a lot of mass. In general, the kinetic energy of a roller coaster rider is at a maximum when the rider reaches a minimum height.
Momentum and Force
Sportscasters often refer to the momentum of a team, and newscasters refer to an election where one candidate has greater momentum Both situations describe a competition where one side is moving toward victory and would be difficult to stop. It seems appropriate to borrow this term form physics, because momentum is a property of movement. It takes longer to stop something from moving when it has a lot of momentum. The physics concept of momentum is closely related to Newton’s laws of motion. Momentum (p) is defined as the product of the mass (m) of an object and its velocity (v)
Momentum = mass x velocity
Or
p=mv
Balanced and Unbalanced Forces
A balanced force results whenever two or more forces act upon an object in such a way as to exactly counteract each other. As you sit in your seat at this moment, the seat pushes upward with a force equal in strength and opposite in direction to the force of gravity. These two forces are said to balance each other, causing you to remain at rest. If the seat is suddenly pulled out from under you, then you experience an unbalanced force. There is no longer an upward seat force to balance the downward pull of gravity, so you accelerate to the ground.
Centripetal force
Motion along a curve or through a circle is always caused by a centripetal force. This is a force that pushes an object in an inward or center seeking direction. The moon orbits the earth in a circular motion because a force of gravity pulls on the moon in an inward direction toward the center of its orbit. In a roller coaster loop, riders are pushed inwards toward the center of the loop by forces resulting from the car seat (at the loop's bottom) and by gravity (at the loop's top).
When you release the string, the centripetal force ceases, and the ball follows its natural straight-line path. The apparent (not a real force) outward force you feel is called a centrifugal force
Work
You learned earlier that the term force has a special meaning in science that is different from your everyday concept of force. In everyday use, you use the term in a variety of associations such as police force, economic force, or the force of an argument. A more precise scientific definition of force was developed from Newton's laws of motion-a force is a result of an interaction that is capable of changing the state of motion of an object. The word work represents a concept that has a special meaning in science that is different from your everyday concept. In everyday use, work is associated with a task to be accomplished or the time spent in performing the task. You also probably associate physical work, such as lifting or moving boxes, with how tired you be- come from the effort. The scientific definition of work is not concerned with tasks, time, or how tired you become from doing a task. It is concerned with the application of a force to an object and the distance the object moves as a result of the force. The work done on the object is defined as the magnitude of the applied force multiplied by the parallel distance through which the force acts:
work = force x distance
W = Fd
Work, in the scientific sense, is the product of a force and the distance an object moves as a result of the force. There are two important considerations to remember about this definition:
(1) something must move whenever work is done, and
(2) the movement must be in the same direction as the direction of the force. When you move a book to a higher shelf in a bookcase you are doing work on the book. You apply a vertically upward force equal to the weight of the book as you move it in the same direction as the direction of the applied force. The work done on the book can therefore be calculated by multiplying the weight of the book by the distance it was moved. If you simply stand there holding the book, you are doing no work on the book. Your arm may become tired from holding the book, since you must apply a vertically upward force equal to the weight of the book. However, this force is not acting through a distance, since the book is not moving. Only a force that results in motion in the same direction results in work.
Describing Motion with Words
Scalars and Vectors
Distance and Displacement
Speed
Velocity
Acceleration
Kinematics is the science of describing the motion of objects using words, diagrams, numbers, graphs, and equations. The goal of any study of kinematics is to develop sophisticated models which serve in describing (and ultimately, explaining) the motion of real-world objects.
Much of our lives are spent in motion, travelling from one place to another. We measure motion by the rate of change that occurs in a certain amount of time.
Scalars and Vectors
Physics is a mathematical science - that is, the underlying concepts and principles have a mathematical basis. The motion of objects can be described by words - words such as distance, displacement, speed, velocity, and acceleration. These mathematical quantities that are used to describe the motion of objects can be divided into two categories. The quantity is either a vector or a scalar. These two categories can be distinguished from one another by their distinct definitions:
Scalars are quantities with only magnitude
Vectors are quantities having both a magnitude and a direction.
Distance and Displacement
The definition of displacement is the change in an object's position. Describing the change in an object's position entails two things: 1) How far did the object go? 2) And in what direction? The direction part is easy. In one-dimensional motion there are only two directions; the positive direction and the negative direction. If the object ends up in the positive direction from where it started it, its displacement is positive. If it ends up in the negative direction from where it started, its displacement is negative.
The only thing that matters when you are talking about displacement is where the object started its motion and where it ended it, NOT where it went in between.
Distance is a scalar measure of the interval between two locations measured along the actual path connecting them. Displacement is a vector measure of the interval between two locations measured long the shortest path connecting them. For example, as illustrated in figure 1, a man rides his bicycle from Manhattan to New Jersey traveling over the Hudson River.
Getting there is a three step process.
1. Follow the Hudson River 8.2 km upriver.
2. Cross using the George Washington
Bridge bike path - 1.8 km between anchorages. 3 Reverse direction and head downriver for 4.5 km.
The distance traveled is 14.5 km, but the resulting displacement is a mere 3.7 km north. The end of this journey is actually visible from the start. Maybe he should buy a canoe.
There is an easy way to mathematically calculate the displacement of an object during any motion. Get the position (on the axis) of the object at the beginning of its motion and the position (on the same axis) of the object at the end of its motion. Be sure to include the positive or negative sign with the position. Then subtract the starting position (the initial position, represented as xi) from the ending position (the final position, represented as xf). This gives you the displacement, including its direction (positive or negative, based on the sign of the answer). Δx =xf-xi
Speed
Speed is a scalar quantity that refers to "how fast an object is moving. Speed is the distance an object travels in a certain amount of time. The speed of an object is constant when the object travels in equal distances in equal periods of time. We define the average speed as the total distance, s, traveled during a particular time divided by the time interval, t. To calculate speed you simply need to divide the distance traveled by the time it took to complete the trip. If the average speed is the same for all parts of travel, then the speed is constant. However, since a moving object often changes its speed during its motion, it is common to distinguish between the average speed and the instantaneous speed. The distinction is as follows:
Instantaneous Speed - speed at any given instant in time
Average Speed - average of all instantaneous speeds, found simply by a distance/time ratio.
Velocity
Velocity is a vector quantity that refers to "the rate at which an object changes its position." It is the displacement or position change during a particular time interval. The task of describing the direction of the velocity vector is easy! The direction of the velocity vector is simply the direction that an object is moving. Average velocity is computed using the equation:
When determining the velocity of an object, the direction must be known. It would not be sufficient to say an object’s velocity is 55 miles/hr. The object's velocity would need to include a direction, 55 miles/hr, east.
What is the difference between
Speed and Velocity?
To specify a velocity, we must give both its size or magnitude (how fast) and its direction (north, south, east, west up, down, right, and left) If you say that an object is moving 15m/s, you have stated its speed. If you say that the object is moving due west at 15m/s, you have indicated its velocity.
As can be seen in Figure 2, the changes in the velocity of the car were produced by forces acting upon the car. The most important force involved in changing the velocity of the car is the frictional force exerted on the tires of the cart by the road surface. A force is required to change either the size or the direction of the velocity. If no net force were acting in the car, it would continue to move at a constant speed in a straight line.
What is a Vector
Has stated previously, a vector is a quantity for which both size and direction are important. Many of the quantities used in describing motion are vector quantities such as velocity, acceleration, force and momentum to name a few.
The length of the arrow is drawn proportional to its size. In other words, the larger the velocity, the longer the arrow will be (fig. 4).
Figure 4 The length of the arrow shows the size of the velocity vector
Acceleration
Acceleration is a familiar idea. We use the term in speaking of the acceleration of a car driving away from a traffic light, or the acceleration of a running back in football. We feel the effects of acceleration on our bodies when a car’s velocity changes rapidly and even mores strikingly when an elevator lurches upward, leaving our stomachs slightly behind. These are all forms of accelerations.
Understanding acceleration is crucial to the study of motion. Acceleration is a vector quantity that is defined as:
“the rate at which an object changes its velocity."
Where a is the acceleration, v1 is the starting velocity, v2 is the ending velocity, and t is the time
An object is accelerating if it is changing its velocity. For example, if a car is headed due north at 10m/sec and you step on the accelerator and increase its velocity to 20 m/sec in 5 sec, you can compute the rate of acceleration as follows:
The final result is read as 2 meters per second per second or 2 meters per second squared. (2m/s2). Assuming your acceleration is uniform, you may interpret the calculation as, the car started out at 10m/sec, but one second later the car increased to 12m/sec; 2 seconds later, 14m/sec; 3 seconds later, 16m/sec; 4 seconds later, 18m/sec; and 5 seconds later, 20m/sec. Therefore, the car increased its velocity 2 meters per second, every second.
The data in the table above is representative of an object accelerating - the velocity is changing with respect to time. In fact, the velocity is changing by a constant amount, 10 m/s in each second of time. Anytime an object's velocity is changing, that object is said to be accelerating
Sometimes an accelerating object will change its velocity by the same amount each second. As mentioned in the above paragraph, the data above shows an object changing its velocity by 10 m/s in each consecutive second. This is referred to as a constant acceleration since the velocity is changing by a constant amount each second. An object with a constant acceleration should not be confused with an object with a constant velocity. Don't be fooled! If an object is changing its velocity - whether by a steady or a varying amount - then it is an accelerating object. In addition, an object with a constant velocity is not accelerating. The data tables below depict motions of objects with a constant acceleration and a changing acceleration. Note that each object has a changing velocity.
Describing Motion with Graphs
The Meaning of Shape for Velocity vs. Time Graph
The velocity vs. time graphs for the two types of motion - constant velocity and changing velocity (acceleration) - can be summarized as follows.
Positive Velocity
Positive Velocity
Zero Acceleration
Positive Acceleration
The shapes of the velocity vs. time graphs for these two basic types of motion - constant velocity motion and accelerated motion (i.e., changing velocity) - reveal an important principle. The principle is that the slope of the line on a velocity-time graph reveals useful information about the acceleration of the object. If the acceleration is zero, then the slope is zero (i.e., a horizontal line). If the acceleration is positive, then the slope is positive (i.e., an upward sloping line). If the acceleration is negative, then the slope is negative (i.e., a downward sloping line). This very principle can be extended to any motion conceivable. The slope of a velocity-time graph reveals information about the object's acceleration. But how can one tell whether the object is moving in the positive direction (positive velocity) or in the negative direction (negative velocity)? And how can one tell if the object is speeding up or slowing down?
Since the graph is a velocity-time graph, the velocity would be positive whenever the line lies in the positive region (above the x-axis) of the graph. Similarly, the velocity would be negative whenever the line lies in the negative region (below the x-axis) of the graph. Finally, if the line crosses over the x-axis from the positive region to the negative region of the graph (or vice versa) then the object has changed its directions.
Now how can one tell if the object is speeding up or slowing down? Speeding up means that the magnitude (the value) of the velocity is getting larger. For instance, an object with a velocity changing from +3 m/s to + 9 m/s is speeding up. Similarly, an object with a velocity changing from -3 m/s to –9 m/s is also speeding up. In each case, the magnitude of the velocity (the number itself, not the sign) is increasing; the speed is getting bigger. Given this fact, one would believe that an object is speeding up if the line on a velocity-time graph is changing from near the 0-velocity point to a location further away from the 0-velocity point. If the line is moving away from the x-axis (the 0-velocity point) then the object is speeding up, and if the line is moving towards the x-axis, then the object is slowing down.
Accelerating object are changing their velocity
The general RULE OF THUMB is:
If an object is slowing down, then its acceleration is in the opposite direction of its motion. This RULE OF THUMB can be applied to determine whether the sign of acceleration of an object is positive or negative, right or left, up or down, etc. Consider the two data tables below. In each case, the acceleration of the object is in the positive "+" direction. In Example A, the object is moving in the positive direction (i.e., has a positive velocity) and is speeding up. When an object is speeding up, the acceleration is in the same direction as the velocity. Thus, this object has a positive acceleration. In Example B, the object is moving in the negative direction (i.e., has a negative velocity) and is slowing down. According to our RULE OF THUMB, when an object is slowing down, the acceleration is in the opposite direction as the velocity. Thus, this object also has a positive acceleration.
Example A
Positive direction speeding up
Example B
Negative direction slowing down
Time
(s)
Velocity
(m/s)
Time
(s)
Velocity (m/s)
0
0
0
-8
1
2
1
-6
2
4
2
-4
3
6
3
-2
4
8
4
0
Both are examples of positive acceleration
This same RULE OF THUMB can be applied to the motion of the objects represented in the two data tables below. In each case, the acceleration of the object is in the negative (-) direction. In Example C, the object is moving in the positive direction (i.e., has a positive velocity) and is slowing down. According to our RULE OF THUMB, when an object is slowing down, the acceleration is in the apposite direction as the velocity. Thus, this object has a negative acceleration. In Example D, the object is moving in the negative direction (i.e., has a negative velocity) and is speeding up. When an object is speeding up, the acceleration is in the same direction as the velocity. Thus, this object also has a negative acceleration. Example C
Positive direction slowing down
Example D
Negative direction speeding up
Time
(s)
Velocity
(m/s)
Time
(s)
Velocity (m/s)
0
8
0
0
1
6
1
-2
2
4
2
-4
3
2
3
-6
4
0
4
-8
Both are examples of negative acceleration
The slope of a velocity-time graph reveals information about the object's acceleration. But how can one tell whether the object is moving in the positive direction (positive velocity) or in the negative direction (negative velocity)? And how can one tell if the object is speeding up or slowing down?
Since the graph is a velocity-time graph, the velocity would be positive whenever the line lies in the positive region (above the x-axis) of the graph. Similarly, the velocity would be negative whenever the line lies in the negative region (below the x-axis) of the graph. And finally, if a line crosses over the x-axis from the positive region to the negative region of the graph (or vice versa), then the object has changed directions.
Now how can one tell if the object is speeding up or slowing down? Speeding up means that the magnitude (the value) of the velocity is getting larger. For instance, an object with a velocity changing from +3 m/s to + 9 m/s is speeding up. Similarly, an object with a velocity changing from -3 m/s to –9 m/s is also speeding up. In each case, the magnitude of the velocity (the number itself, not the sign) is increasing; the speed is getting bigger. Given this fact, one would believe that an object is speeding up if the line on a velocity-time graph is changing from near the 0-velocity point to a location further away from the 0-velocity point. That is, if the line is moving away from the x-axis (the 0-velocity point), then the object is speeding up. Conversely, if the line is moving towards the x-axis, then the object is slowing down.
Force and Acceleration
Is there a relationship between force and acceleration? An applied force is what causes an object to accelerate. The force along with the mass of an object determines the degree of response (acceleration) that the object experiences as a result of the force, but remember acceleration does not cause a force.
Mass
In the formula, a = F/m, it is also easier to determine the effect of mass on the amount of acceleration that an object experiences as a result of an applied force. Since mass (m) appears on the bottom, we know that mass is inversely related to acceleration. What this means is that if the same amount of force is applied to two objects, the object with the larger mass will experience a smaller amount of acceleration (or response to the force).
Forces Are Additive
For this discussion, consider motion in one dimension. If you and a friend push a box with the same amount of force but in opposite directions, the box will not move, the forces cancel one another. As an example to further illustrate the point, you already know that it is easier to push a stalled car when two people are pushing it instead of just one person. Well, this is obvious you might say, but it provides an example of the additive nature of forces. Let us assume that it takes 50 Newtons of force to move a stalled car. If you were just pushing it by yourself, you would have to provide all of this force by yourself. However, if two people were pushing the car, your combined total would have to be 50 Newtons. Therefore, you could push 20 Newtons, and your friend could push 30 Newtons so that the total force would still be 50 Newtons.
Direction
We know that acceleration involves a direction because the definition of acceleration is a change in velocity or a change in direction over time. Likewise, force has a direction, namely the direction in which it is applied. Therefore, the direction in which the force is applied is also the direction in which the object accelerates. This should be obvious because the acceleration that an object experiences is the direct response to the applied force. Therefore, a force causes an object to change its velocity. Objects in nature tend to maintain a constant velocity, unless they have contact with a nonzero net force.
Acceleration in Detail
If you are still having difficulty determining the direction of the acceleration, it may be helpful to figure out the direction of the force first. Once the direction of the force is known, the direction of the acceleration will be in the same direction. For example, consider an object that is moving to the right and speeding up. Is the object accelerating? If so, in what direction is the object's acceleration?
Well, the first question is easily answered. All we have to do is recall the definition of acceleration. Anytime an object's velocity is changing; the object is also accelerating. Since velocity involves both direction and speed, if either of these is changing, then the velocity is changing. In this case, the speed is changing (increasing) while the direction is remaining constant; therefore the object's velocity is changing so the object is accelerating.
Now we know the object is accelerating, but in which direction is the force? If the object is moving to the right and speeding up then the object's velocity is changing (object is accelerating), so there must be a force. This is because of Newton's 1st Law of Motion; if an object is not moving at a constant velocity, then there must be a net force acting on that object. Now that we have established there is a net force acting on the object, then determine the direction that the force is exerted. Picture a car in neutral moving to the right. Which direction would you have to push the car to get it to speed up while moving to the right? The answer is that you would have to push to the right. Since we now know that the direction of the force is to the right and that the direction of the force and acceleration must be in the same direction, then the direction of the acceleration is to the right. This example involves an object that is "accelerating" instead of "decelerating".
The first conclusion that can be drawn is that, when the force is in the same direction as the velocity, the object will speed up in the same direction as the velocity. In other words, when the direction of the force is in the same direction as the velocity, the object "accelerates". In this example, the direction of the velocity is to the right because the object was moving to the right. This is in the same direction as the force, which was pointing to the right, as well.
The second conclusion that one can draw from this example is that an object speeds up in the direction of its velocity when the direction of its acceleration and its velocity are the same. Alternatively, one could say that, when the direction of acceleration is the same as the direction of an object's velocity, the object will "accelerate". This obviously follows from the first conclusion because we know that the direction of the force and acceleration are the same.
Example 2
Consider an object moving to the right and slowing down. Is the object accelerating? If so, in which direction is the object's acceleration?
Once again, the first question is easily answered. Since the velocity is changing in this example, the object is accelerating. Because the speed is changing (decreasing), the velocity is changing; therefore the object must be accelerating.
To answer the second question, it is, once again, easier to think of the direction of the force. Rephrase the questions above, if an object is moving to the right and slowing down, is there a net force present? If so, in which direction does this net force have to be in order to slow down the object while it is moving to the right?
Well, since the object's velocity is not constant, there must be a net force acting on that object. Consider a car moving to the right in neutral. In which direction would the force have to act in order for it to slow down? The answer is that the force would have to be to the left, or in the opposite direction. If we know the direction of the force, then we know the direction of the acceleration, as well, because they are always in the same direction. Therefore, the direction of the acceleration points to the left.
In summary, when the net force and the velocity are in opposite directions, as is true in this example, the object slows down or "decelerates". Likewise, we can say that an object "decelerates" when the direction of its acceleration is opposite to that of its velocity. Therefore, when we say an object is "decelerating", what we really are saying is that the direction of its acceleration is opposite to the direction of its velocity.
If the force remains in the same direction after the object's velocity drops to zero, the object will begin to speed up to the left. After the velocity drops to zero, the force and the velocity are in the same direction.
Therefore, if a net force is applied in a direction opposite to the object's velocity, the object will initially continue moving in the same direction while slowing down to zero. Then the object will begin to speed up in the direction of the force, which is opposite to the direction in which it was initially moving. In other words, when a net force is applied opposite to the velocity of an object, the object will initially "decelerate" to zero (while moving in the same direction that it was initially moving) and then it will begin to "accelerate" in the direction opposite to the initial direction that the object was moving in.
In summary:
1. When the net force is in the same direction as the object's velocity, the object will continue to move in the same direction, while its speed will increase. In other words, the object will "accelerate".
2. When the net force is opposite to the direction of the object's velocity, the object will initially continue to move in the same direction while slowing down. After the object reaches zero velocity, it will begin to speed up in the direction opposite to the direction of its initial velocity. In other words, the object will initially "decelerate" while moving in the same direction. Once the object "decelerates" to zero, (stops) it will then "accelerate" in the direction opposite to its initial velocity.
3. You might have wondered why I didn't consider the case of zero net force. In fact, we have already considered this possibility. This possibility is exactly what Newton's 1st Law of Motion describes. If an object is subjected to zero net force, then it will continue to move at a constant velocity, meaning that it will continue to move in a straight line at a constant speed. Furthermore, it will continue to move until it is subject to a nonzero net force.
Free Fall and the Acceleration of Gravity
A free-falling object is an object that is falling under the sole influence of gravity. It is a remarkable fact, first discovered over 300 years ago by Galileo that objects in free fall motion descend at a constant acceleration. During each second of the fall, the object gains 9.8 m/s2 (32 ft/s2) in velocity. This means that under free fall all objects have the same constant, acceleration equal to 9.8 m/s2. The acceleration of a free-falling object is known as acceleration due to gravity, and is denoted by the symbol g.
If you dropped, an object is the falling object accelerating. Think for a moment about whether the velocity is changing. Before you release an object, its velocity is zero, but the instant after the object is released, the velocity has some value different from zero. There has been a change in velocity. If the velocity is changing, there is acceleration.
Things happen so rapidly that it is difficult, just from watching the fall, to say much about the acceleration. It does appear to be large, because the velocity increases rapidly. Does the object reach a large velocity instantly, or does the acceleration occur more uniformly?
To answer this question, we must slow the motion down somehow so that our eyes and brains can keep up with what is happening. There are several ways to slow down the action. One is by using a stroboscope, a rapidly blinking light whose flashes occur at regular intervals in time. In Figure 5, a photograph taken using a stroboscope is used to illuminate an object as it falls. The position of the object is pinpointed every time the light flashes. If you look closely you will notice the distance covered is successive time intervals increases regularly. The time intervals between successive positions of the ball are all equal.
Figure 5
Time
Distance
Velocity
Time
Distance
Velocity
1
0
1.2cm
24cm/s
5
0.20s
19.7cm
218cm/s
2
0.05s
4.8cm
72cm/s
6
0.25s
30.6cm
268cm/s
3
0.10s
11.0cm
124cm/s
7
0.30s
44.0cm
320cm/s
4
0.15s
19.7cm
174cm/s
8
0.35s
60.0cm
368cm/s
You could verify the other values shown in the third column of table 3.1 by doing similar computations. It is clear in table above that the velocity values steadily increase. To see that velocity is increasing at a constant rate, we can plot velocity against time (fig. 3.4). Notice that each velocity data point is plotted at the midpoint between the two times (or flashes) from which it was computed. This is because these values represent the average velocity for the short time intervals between flashes
Notice that the slope of the line is constant in the graph above. The velocity values all fall approximately on a constant-slope straight line. Since acceleration is the slope of the velocity-versus-time graph, the acceleration must also be constant. The velocity increases uniformly with time.
To find the value of the acceleration, we choose two velocity values that lie on the straight line and calculate how rapidly the velocity is changing. For example, the last velocity value, 368 cm/s, and the second value, 72 cm/s, are separated by a time interval corresponding to 6 flashes or 0.30 second. The increase in velocity Δv is found by subtracting 72 cm/s from 368 cm/s, obtaining 296 cm/s. To find the acceleration, we divide this change in velocity by the time interval
cm/s2 = 9.8 m/s2
This result gives us the acceleration due to gravity for objects falling near the earth’s surface. Its value actually varies slightly from point to point on the earth’s surface because of differences in altitude and other effects. This acceleration is used so often that it is given its own symbol
g where g =9.8 m/s2
Called the gravitational acceleration or acceleration due to gravity, it is valid only near the surface of the earth and thus is not a fundamental constant.
The diagram below shows the position of a free-falling object every second. The fact that the distance, which the object travels every second, is increasing is a sure sign that the object is speeding up as it falls downward. Remember that if an object travels downward and speeds up, then its acceleration is downward. g = 9.8 m/s/s, downward
Figure 6
Figure 7 Free fall acceleration: Graphs of how the position (a), velocity (b), and acceleration (c) of objects under free fall motion change in time.
Acceleration Due to Gravity
Galileo predicted that heavy objects and light ones would fall at the same rate. The reason for this is simple. Suppose the coin has 50 times as much mass as the feather. This means that the earth pulls 50 times as hard on the coin as it does on the feather. You might think this would cause the coin to fall faster. But because of the coin's greater mass, it's also much harder to accelerate the coin than the feather - 50 times harder, in fact! The two effects exactly cancel out, and the two objects therefore fall with the same acceleration.
Near the surface of the earth, gravity is a downward acceleration. Over time, it causes all [unsupported] objects to pick up a downward speed.
Since gravity is a persistent acceleration, falling objects keep falling faster and faster.
Gravity accelerates objects towards the earth at 9.8 meters per second per second
This means each second it falls, it will be traveling 9.8 meters per second faster
If something is dropped from 227.00 meters (744.75 feet or 0.14 miles), it will hit the ground in 6.81 seconds
It will be traveling at 66.70 meters (149.21 mph) per second when it hits the ground
Falling with Air Resistance
This rule holds true only if gravity is the only force acting on the two objects. As an object falls through air, it usually encounters some degree of air resistance. Air resistance is the result of collisions of the object's leading surface with air molecules. The actual amount of air resistance encountered by the object is dependent upon a variety of factors: the speed of the object and the cross-sectional area of the object.
Increased speeds result in an increased amount of air resistance. Increased cross-sectional areas result in an increased amount of air resistance.
As an object falls, it picks up speed. The increase in speed leads to an increase in the amount of air resistance. Eventually, the force of air resistance becomes large enough to balances the force of gravity. At this instant in time, the net force is 0 Newtons; the object will stop accelerating. The object is said to have "reached a terminal velocity." As stated above, the amount of air resistance depends upon the speed of the object. A falling object will continue to accelerate to higher speeds until they encounter an amount of air resistance that is equal to their weight. Since the 150-kg skydiver weighs more (experiences a greater force of gravity), he will accelerate to higher speeds before reaching a terminal velocity. Thus, more massive objects fall faster than less massive objects because they are acted upon by a larger force of gravity; for this reason, they accelerate to higher speeds until the air resistance force equals the gravity force.
Representing Free Fall by Graphs
Free-falling motion can be represented using two basic graphs; position vs. time and velocity vs. time graphs.
A position vs. time graph for a free-falling object is shown below.
Observe that the line on the graph curves. As learned earlier, a curved line on a position vs. time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration g =10 m/s/s = approximate value), it would be expected that its position-time graph would be curved. A further look at the position-time graph reveals that the object starts with a small velocity (slow) and finishes with a large velocity (fast). Since the slope of any position vs. time graph is the velocity of the object the small initial slope indicates a small initial velocity and the large final slope indicates a large final velocity. Finally, the negative slope of the line indicates a negative (i.e., downward) velocity.
A velocity vs. time graph for a free-falling object is shown below.
Observe that the line on the graph is a straight, diagonal line. As learned earlier, a diagonal line on a velocity vs. time graph signifies an accelerated motion. Since a free-falling object is undergoing an acceleration (g = 10 m/s/s), it would be expected that its velocity-time graph would be diagonal. A further look at the velocity-time graph reveals that the object starts with a zero velocity (as read from the graph) and finishes with a large, negative velocity; that is, the object is moving in the negative direction and speeding up. An object which is moving in the negative direction and speeding up is said to have a negative acceleration (if necessary, review the vector nature of acceleration). Since the slope of any velocity vs. time graph is the acceleration of the object, the constant, negative slope indicates a constant, negative acceleration. This analysis of the slope on the graph is consistent with the motion of a free-falling object - an object moving with a constant acceleration of 10 m/s/s in the downward direction.
Newton's Laws of Motion
First Law of Motion An object that is at rest will remain at rest unless a nonzero net (or total) force is exerted on it.
Newton's First Law, is also called the Law of inertia. Inertia is a tendency of an object to resist change in its state of motion. More massive objects have more inertia; that is, they have more tendencies to resist changes in the way they are moving. An elephant has a lot of inertia, for example. If it is at rest, it offers a large resistance to changes in its state of rest, and so it is difficult to move an elephant. On the other hand, a pencil has a small amount of inertia. It is easy to move a pencil from its state of rest. More massive objects have more inertia and thus require more force in order to change their state of motion. This one is easy to believe and sounds intuitive enough. However, the next part of the first law might sound less plausible. Simply stated, an object moving at a constant velocity will continue to move at a constant velocity (moving at a constant speed and in a straight line) unless a nonzero net (or total) force acts upon the object. Recall that constant velocity means that the object is moving at a constant speed and in a constant direction. You might think about something you’ve seen in the world that contradicts this statement. For instance, think of a car rolling in a straight line while in neutral. If the statement above is true, then the car should be able to roll in a straight line at a constant speed forever. This is obviously not true in real life because we know the car will eventually come to a stop. This certainly seems to contradict Newton's first law, or does it? Newton's first law states that an object moving at a constant velocity will continue moving in that fashion unless a nonzero net force acts upon that object. It is true that the car would continue to move in a straight line at a constant speed if there was no net force acting on the car. However, is there really no net force acting on the car? The force responsible for slowing down the car is friction. Therefore, the above observation, a car slowing down while in neutral, is not inconsistent with Newton's first law of motion.
In summary, objects like to move in straight lines, at constant speeds unless a force acts upon them. In fact, when a force acts on an object, the force causes the object to change its velocity. In other words, forces cause objects to accelerate.
Second Law of Motion
The acceleration of an object is directly proportional to the magnitude of the imposed force and inversely proportional to the mass of the object. The acceleration is in the same direction as that of the imposed force.
Simply stated, a force causes an object to accelerate. Whenever you see an object accelerating, there must be an external force acting on the object because, as stated in Newton's first law, objects move at a constant velocity unless acted upon by an outside force.
Mathematically, Newton's second law of motion can be expressed by the following formula: A = F/M where
A = acceleration,
F = force, and
M = mass.
What this formula tells us is that force causes an object to accelerate. However, it also tells us that the acceleration an object feels, in response to an applied force, does not solely depend on the amount of force applied. It also depends on the mass or inertia of that object. It also tells us that the more mass an object has, the less it accelerates in response to an applied force. This makes intuitive sense. For instance, if I apply the same force to a cotton ball and a book, the cotton ball would experience a greater acceleration than the book because the book has much more mass or inertia. Therefore, an object with a greater mass has a better tendency to resist a change in its motion when an external force is applied to that object. In other words, we say that the book has more inertia than the cotton ball.
Third Law of Motion
Newton's third law states that whenever a force is exerted, an equal and opposite force arises in reaction to this force. In other words, every force has an equal and opposite reaction force.
For example, when you push on a wall, the wall will also push back on you with an equal and opposite force. If this is true, why don’t you move backwards if the wall is pushing you backwards? The reason you do not move backwards when you push against a wall is that static friction is pushing you back with an equal amount of force to the right so that you do not move anywhere. In the above example, static friction would be pushing the person to the right with five Newton’s (5 N) of force, so that the person would experience zero net force, hence the person does not move. Therefore, if Newton's third law is true, and the wall pushes back on us just as hard as we push back on it,
Amusement Park Physics
When you enter an amusement park, you are walking into the world of physics. Some of us have found the nerve to ride a roller coaster or perhaps drop from a 35-story tower in free fall. Amusement park rides are designed to provide the rider with spine-chilling experiences. The laws of physics govern the extent of the thrill we experience. If we examine the physics involved in a roller coaster ride, the following properties can be observed:
Free-fall rides are made up of three distinct parts. During the first part of the ride, force is applied to the car to lift it to the top of the free-fall tower. As the rider climbs up a steep incline, they feel pressed against their seats; this provides a sense of weight. However, at the top of the tower, a 100-pound rider would feel 100 pounds of force from the seat pushing an external force upon their body. The rider feels their normal weight. The amount of force that must be applied depends on the mass of the car and its passengers. Motors apply the force, and there is a built-in safety allowance for variations in the mass of the riders. From the moment you begin climbing up the incline, you are building up potential energy, which is the stored energy that has the potential to be converted into kinetic energy. Potential energy is the energy possessed by an object because of its height above the ground. The amount of potential energy possessed by an object depends on its mass and its height. A roller coaster car is initially hauled by a motor and chain system to the top of a tall hill, giving it a large quantity of potential energy.
Suddenly the cart drops, plunging down the incline, accelerating toward the ground under the sole influence of gravity. As the rider free falls from the tower, they fall at the same rate as their surroundings-in this case, their seat. It is as though the seat has fallen out from under them; the rider no longer feels the external force of the seat and subsequently has a brief sensation of weightlessness. They have not lost any weight, but feels as though they have because of the absence of the seat force. In this context, weightlessness is a sensation and not an actual change in weight.
As the cart free falls, it is accelerating and is measured in g’s. A g is a unit of acceleration equal to the acceleration caused by gravity. Gravity causes free-falling objects on the earth to change their speeds at rates of about 10 m/s each second. That would be equivalent to a change in speed of 32 feet per second in each consecutive second. If an object is said to experience 3 g's of acceleration, then the object is changing its speed at a rate of about 30 m/s every second.
Once you begin your descent downwards, that energy is being converted into kinetic energy which is the energy of motion. So all of that potential energy that was stored up became kinetic energy. Kinetic energy is the energy possessed by an object because of its motion. All moving objects have kinetic energy. The amount of kinetic energy depends upon the mass and speed of the object. A roller coaster car has a lot of kinetic energy if it is moving fast and has a lot of mass. In general, the kinetic energy of a roller coaster rider is at a maximum when the rider reaches a minimum height.
Momentum and Force
Sportscasters often refer to the momentum of a team, and newscasters refer to an election where one candidate has greater momentum Both situations describe a competition where one side is moving toward victory and would be difficult to stop. It seems appropriate to borrow this term form physics, because momentum is a property of movement. It takes longer to stop something from moving when it has a lot of momentum. The physics concept of momentum is closely related to Newton’s laws of motion. Momentum (p) is defined as the product of the mass (m) of an object and its velocity (v)
Momentum = mass x velocity
Or
p=mv
Balanced and Unbalanced Forces
A balanced force results whenever two or more forces act upon an object in such a way as to exactly counteract each other. As you sit in your seat at this moment, the seat pushes upward with a force equal in strength and opposite in direction to the force of gravity. These two forces are said to balance each other, causing you to remain at rest. If the seat is suddenly pulled out from under you, then you experience an unbalanced force. There is no longer an upward seat force to balance the downward pull of gravity, so you accelerate to the ground.
Centripetal force
Motion along a curve or through a circle is always caused by a centripetal force. This is a force that pushes an object in an inward or center seeking direction. The moon orbits the earth in a circular motion because a force of gravity pulls on the moon in an inward direction toward the center of its orbit. In a roller coaster loop, riders are pushed inwards toward the center of the loop by forces resulting from the car seat (at the loop's bottom) and by gravity (at the loop's top).
When you release the string, the centripetal force ceases, and the ball follows its natural straight-line path. The apparent (not a real force) outward force you feel is called a centrifugal force
Work
You learned earlier that the term force has a special meaning in science that is different from your everyday concept of force. In everyday use, you use the term in a variety of associations such as police force, economic force, or the force of an argument. A more precise scientific definition of force was developed from Newton's laws of motion-a force is a result of an interaction that is capable of changing the state of motion of an object. The word work represents a concept that has a special meaning in science that is different from your everyday concept. In everyday use, work is associated with a task to be accomplished or the time spent in performing the task. You also probably associate physical work, such as lifting or moving boxes, with how tired you be- come from the effort. The scientific definition of work is not concerned with tasks, time, or how tired you become from doing a task. It is concerned with the application of a force to an object and the distance the object moves as a result of the force. The work done on the object is defined as the magnitude of the applied force multiplied by the parallel distance through which the force acts:
work = force x distance
W = Fd
Work, in the scientific sense, is the product of a force and the distance an object moves as a result of the force. There are two important considerations to remember about this definition:
(1) something must move whenever work is done, and
(2) the movement must be in the same direction as the direction of the force. When you move a book to a higher shelf in a bookcase you are doing work on the book. You apply a vertically upward force equal to the weight of the book as you move it in the same direction as the direction of the applied force. The work done on the book can therefore be calculated by multiplying the weight of the book by the distance it was moved. If you simply stand there holding the book, you are doing no work on the book. Your arm may become tired from holding the book, since you must apply a vertically upward force equal to the weight of the book. However, this force is not acting through a distance, since the book is not moving. Only a force that results in motion in the same direction results in work.
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