Kinematics: displacement, velocity and acceleration Cambridge IGCSE Additional Mathematics revision
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In plain words
Position, velocity and acceleration are a ladder. Differentiate to go down a rung: position to velocity to acceleration. Integrate to climb back up. That is the whole of this topic.
7 things to know
- Displacement s is the position of a particle measured from a fixed point. Velocity v is the rate of change of displacement: v = ds/dt.
- Acceleration a is the rate of change of velocity: a = dv/dt.
- Going the other way: v is the integral of a, and s is the integral of v, with respect to t. Each brings a constant, found from given information such as "at rest when t = 0".
- A particle is at rest when v = 0. It changes direction when v changes sign.
- Velocity is greatest or least when a = 0.
- Distance travelled is not the same as displacement. If the particle turns round, find the displacement at each time when v = 0 and add up the separate distances.
- On graphs: the gradient of a displacement–time graph is the velocity. The gradient of a velocity–time graph is the acceleration, and the area under it is the displacement.
Worked example
A particle moves in a straight line so that s = t³ − 6t² + 9t, where s is in metres and t in seconds. Find when it is at rest, its acceleration at t = 2, and the distance it travels in the first 3 seconds.
- v = ds/dt = 3t² − 12t + 9 = 3(t − 1)(t − 3). It is at rest when t = 1 and when t = 3.
- a = dv/dt = 6t − 12. At t = 2, a = 0.
- s = 0 at t = 0, s = 4 at t = 1, and s = 0 at t = 3. It goes 4 m out and 4 m back: 8 m in total.
Worked example
A particle has acceleration a = 6t − 4. When t = 0, its velocity is 5 and its displacement is 0. Find v and s in terms of t.
- v = 3t² − 4t + c. When t = 0, v = 5, so c = 5: v = 3t² − 4t + 5.
- s = t³ − 2t² + 5t + k. When t = 0, s = 0, so k = 0: s = t³ − 2t² + 5t.
Tips and tricks
- "Instantaneously at rest" means v = 0. "Initially" means t = 0.
- Speed is the size of the velocity. A velocity of −3 is a speed of 3, in the negative direction.
It lands in your notebook with its questions as flashcards.
Kinematics: displacement, velocity and acceleration: 6 questions and answers
These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.
How is velocity found from displacement?
v = ds/dt.
How is displacement found from velocity?
Integration climbs back up the ladder.
What is true when a particle is instantaneously at rest?
At rest means not moving at that instant.
s = t² + 3t. What is the velocity when t = 2?
v = 2t + 3.
v = 6t − 2. What is the acceleration?
a = dv/dt.
What does the area under a velocity–time graph give?
The gradient gives the acceleration.
Quiz
6 questions
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Worksheet
3 questions, 10 marks. Write your answers on paper, then check them.
Kinematics: displacement, velocity and acceleration
Cambridge IGCSE Additional Mathematics 0606 · 10 marks · papermunch.org
Name ______________________________ Date ______________
s = 2t³ − 9t². Find the velocity and the acceleration when t = 2.[3]
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The velocity is −12 and the acceleration is 6. v = 6t² − 18t and a = 12t − 18.
A particle has velocity v = 12t − 3t². Find the times when it is at rest, and its displacement between those times.[4]
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It is at rest when t = 0 and t = 4. The displacement is 32. [6t² − t³] from 0 to 4 gives 96 − 64.
A particle has acceleration a = 4 − 2t, and v = 3 when t = 0. Find its greatest velocity.[3]
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7. v = 4t − t² + 3. The velocity is greatest when a = 0, at t = 2.
Answers: Kinematics: displacement, velocity and acceleration
- 1. The velocity is −12 and the acceleration is 6. v = 6t² − 18t and a = 12t − 18.
- 2. It is at rest when t = 0 and t = 4. The displacement is 32. [6t² − t³] from 0 to 4 gives 96 − 64.
- 3. 7. v = 4t − t² + 3. The velocity is greatest when a = 0, at t = 2.