Chapter 21 Kinematics 21.1 Displacement, Velocity and Acceleration

Kinematics In this lesson, you will look at kinematics of a particle moving along a straight line, using terms such as displacement, velocity and acceleration. You will learn to apply the equations You will learn to find displacement, velocity and acceleration at an instant and how to find the actual distance travelled from the direction of motion of its velocity. Objectives 21.1 Displacement, Velocity and Acceleration

A particle P is moving in a straight line. t seconds after passing a fixed point O, its distance s metres from O, is given by s = 4t – t 2, 0 ≤ t ≤ 4. Kinematics O 2134 AB PPP Initially P is at O. When t = 1, s = 4 – 1 = 3, so P is at A. When t = 2, s = 8 – 4 = 4, so P is at B. When t = 3, s = 12 – 9 = 3, so P is at A. v = 4 – 2 = 2.P moves to the right. v = 4 – 4 = 0.P is stationary. v = 4 – 6 = –2.P moves to the left.

Integrate Acceleration to get Velocity Displacement, s Differentiate Displacemen t to get Velocity Integrate Velocity to get Displacemen t Differentiat e Velocity to get Acceleratio n Kinematics Velocity, v Acceleration, a Summary

A Particle Q moves in a straight line so that its distance, s m, from a fixed point O is given by s = 6t 2 –t 3. Q goes 32 m forward and then 7 m back. Speed is velocity without direction. Kinematics Find the velocity and speed of Q after 5 seconds. (the speed of Q is 15 ms -1 ). Find the distance from O when Q is instantaneously at rest. Q is 32 m from O. Find the total distance travelled in the first 5 seconds. The total distance travelled is 32 + (32 – 25) = 39 m. Example 1

Find expressions for the velocity and acceleration in terms of t. Kinematics Find the velocity of the particle when it is 5 m from O. Find the time t when the acceleration is A particle travels in a straight line so that at time t seconds its displacement, s m, from a fixed point O is given by Example 2

If the initial displacement is zero, find v and s in terms of t. Kinematics Find the time taken to return to O and the acceleration at this instant. A particle moves from rest at point O in a straight line so that at time t seconds its acceleration, a ms -2, is given by Example 3

Kinematics Displacement-time graph s (m) t (s) 60 At rest Constant velocity (−ve) Constant velocity (+ve) Initial displacement = 20 m

Kinematics Velocity-time graph v (m/s) t (s) 60 Constant velocity = zero acceleration Decelerating at a constant rate Accelerating at a constant rate Initial velocity = 20 m/s Area under graph = displacement Accelerating in the opposite direction