# Newtonian Mechanics

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Chapter 2: Kinematics Rectilinear Motion Non-linear Motion a. Define displacement, speed, velocity and acceleration. Distance: Displacement: Speed: Velocity: Total length covered irrespective of the direction of motion. Distance moved in a certain direction Distance travelled per unit time. is defined as the rate of change of displacement, or, displacement per unit time {NOT: displacement over time, nor, displacement per second, nor, rate of change of displacement per unit time} is defined as the rate of change of velocity.

Acceleration: b.

Use graphical methods to represent distance travelled, displacement, speed, velocity and acceleration. Self-explanatory

c.

Find displacement from the area under a velocity-time graph. The area under a velocity-time graph is the change in displacement.

d.

Use the slope of a displacement-time graph to find velocity. The gradient of a displacement-time graph is the {instantaneous} velocity.

e.

Use the slope of a velocity-time graph to find acceleration. The gradient of a velocity-time graph is the acceleration.

f. g.

Derive, from the definitions of velocity and acceleration, equations that represent uniformly accelerated motion in a straight line. Solve problems using equations which represent uniformly accelerated motion in a straight line, including the motion of bodies falling in a uniform gravitational field without acceleration. 1. 2. 3. 4. v = u +a t: s = ½ (u + v) t: 2 2 v = u + 2 a s: 2 s=ut+½at : derived from definition of acceleration: a = (v – u) / t derived from the area under the v-t graph derived from equations (1) and (2) derived from equations (1) and (2)

These equations apply only if the motion takes place along a straight line and the acceleration is constant; {hence, for eg., air resistance must be negligible.} h. Describe qualitatively the motion of bodies falling in a uniform gravitational field with air resistance. Consider a body moving in a uniform gravitational field under 2 different conditions: A WITHOUT AIR RESISTANCE Highest point v t W +ve Moving up Moving down

Assuming negligible air resistance, whether the body is moving up, or at the highest point or moving down, the weight of the body, W, is the only force acting on it, causing it to experience a constant acceleration. Thus, the gradient of the v-t graph is constant throughout its rise and fall. The body is said to undergo free

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fall.

B v

WITH AIR RESISTANCE Highest point gradient = 9.81 t Terminal velocity Moving up Moving down

If air resistance is NOT negligible and if it is projected upwards with the same initial velocity, as the body moves upwards, both air resistance and weight act downwards. Thus its speed will decrease at a rate -2 greater than 9.81 m s . This causes the time taken to reach its maximum height reached to be lower than in the case with no air resistance. The max height reached is also reduced. At the highest point, the body is momentarily at rest; air resistance becomes zero and hence the only force -2 acting on it is the weight. The acceleration is thus 9.81 m s at this point. As a body falls, air resistance opposes its weight. The downward acceleration is thus less than 9.81 m s . As air resistance increases with speed (Topic 5), it eventually equals its weight (but in opposite direction). From then there will be no resultant force acting on the body and it will fall with a constant speed, called the terminal velocity. i. Describe and explain motion due to a uniform velocity in one direction and uniform acceleration in a perpendicular direction. Equations that are used to describe the horizontal and vertical motion x direction (horizontal – axis) s x = ux t s (displacement) 1 2 s x = ux t + 2ax t y direction (vertical – axis) 1 2 s y = uy t + 2 a y t (Note: If projectile ends at same level as the start, then s y = 0) -2

u (initial velocity)...

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