In this experiment you will study the motion of an object in one dimension from a number of points of view. You will demonstrate how the variables of motion are related by differentiation and integration and investigate the relationship between potential and kinetic energy.

Theory

Why Study Motion?

Motion is everywhere in the universe. Only at a temperature of absolute zero is the motion in any body truly absent. If motion exists then so also does energy. To the delight of the modern-day physicist the tools that were invented by Galileo Galilei, Isaac Newton and others 200 years ago to describe motion apply everywhere in the universe, from electrons in our own bodies to the farthest galaxy. The study of motion and of energy is at the heart of physics. This experiment deals with motion of the simplest kind, motion in one dimension or motion in a straight line.

Kinematics and Dynamics

The subject of motion is divided for convenience into the subtopics of kinematics and dynamics. Kinematics is concerned with the aspects of motion that exclude the forces that cause motion. In a manner of speaking, kinematics is focussed on the development of definitions: position, displacement, velocity, acceleration and on the relationships that exist between them. Dynamics widens the study of motion to include the concepts of force and energy.

Definitions

Position Kinematics begins with the idea of position. Suppose that we photograph an object moving to the left along a horizontal path at two instants of time and superimpose the images for study (Figure 1). We examine one image with a ruler and mark off the number of units that separate the object from the ruler’s zero. The zero is a reference or origin at a position of zero units by definition. The position of the object at any another place is, say x units. x is an instantaneous quantity since it applies to a specific clock time—the instant the photograph was taken. Position like length is a basic quantity and is dependent only on the unit used. But position involves direction also. In principle the object could be to our right or to our left. To include the information of direction we use a vector. The magnitude or length of the vector, say r, is r (or perhaps x), while the direction is to the right, meaning the object is to the right of the reference point. We could also agree that, by convention, the sign of x is positive in this particular case. Elapsed Time The two positions of the object in Figure 1 must be described with different vectors and different clock times. The photographs can be said to show two events, an initial “i” event and a final “f” event. There is now an elapsed time between the events equal to the simple difference:

Δt = t f – t i ,

…[1]

(unit seconds, abbreviated s). Keep in mind that the concepts of clock time and elapsed time are different; an elapsed time is the difference between two clock times.

L2-1

L2 Linear Motion

0

rf

clock time tf object ri

displacement Δ r = rf – ri

clock time ti object Δr = v Δt

Figure 1. This drawing illustrates an object moving toward the origin (left) “photographed” at two positions. The corresponding clock times are indicated. Position, displacement and velocity vectors are given different head styles to emphasize their different natures.

Displacement Displacement differs from position. In the elapsed time between the events the object moves from one position to another. The displacement is the difference between the two vectors describing the two positions:

d. Eq[3] then becomes what is known as the instantaneous velocity

dr =v. dt

…[4]

Δ r = rf – ri ,

…[2]

(unit meters, abbreviated m). Displacement, being the difference between two vectors, is also a vector. The displacement is negative in this case (according to our convention) since it points towards the origin. Velocity Average Velocity. Another quantity in kinematics is the average...