Kinematics – Year 12 A level maths worksheet with answers to download as PDF

A lift travels between floors. It accelerates uniformly from rest to 12 m/s in 6 s, travels at 12 m/s for 20 s, then decelerates uniformly to rest in 4 s.

1. Find the acceleration in the first stage and the deceleration in the last stage. [2 marks]
2. Find the distance travelled in each stage. [3 marks]
3. Find the total distance and the average speed for the whole journey. [3 marks]

Use the equations of constant acceleration ("suvat").

1. A particle has \(u=3\) m/s and \(a=2\) m/s\(^2\). Find \(v\) and \(s\) after 5 s. [3 marks]
2. A particle has \(u=5\) m/s, \(v=17\) m/s and \(s=66\) m. Find \(t\) and \(a\). [4 marks]
3. A particle travels 48 m in 4 s with acceleration 1.5 m/s\(^2\). Find the initial speed. [3 marks]

A car travelling at 20 m/s brakes with a constant deceleration of 4 m/s\(^2\).

1. Find the braking distance. [3 marks]
2. Find the time taken to stop. [2 marks]
3. Find the speed after it has travelled 30 m while braking. [3 marks]

A ball is thrown vertically upwards from the ground at 14 m/s. Take \(g=9.8\) m/s\(^2\).

1. Find the maximum height. [3 marks]
2. Find the time to reach the maximum height and the total time in the air. [3 marks]
3. Find the velocity and height after 2 s. [4 marks]

A particle moves in a line with displacement \(s=2t^3-9t^2+12t\) metres at time \(t\) seconds.

1. Find expressions for the velocity and the acceleration. [3 marks]
2. Show that the particle is at rest at \(t=1\) and \(t=2\). [2 marks]
3. Find the displacement at each of these times and the acceleration at \(t=1\). [4 marks]

A particle has acceleration \(a=6t-4\) m/s\(^2\). When \(t=0\) its velocity is 3 m/s and its displacement is 0.

1. Find the velocity at time \(t\). [3 marks]
2. Find the displacement at time \(t\). [3 marks]
3. Find the velocity and displacement when \(t=2\). [2 marks]

A train leaves a station, accelerates uniformly at 0.5 m/s\(^2\) until it reaches 25 m/s, travels at this speed for 4 minutes, then decelerates uniformly at 1 m/s\(^2\) to stop at the next station.

1. Find the time for each of the three stages. [3 marks]
2. Find the distance between the stations. [4 marks]
3. Find the average speed for the whole journey in m/s and km/h. [3 marks]
4. Sketch the velocity-time graph. [2 marks]

A driver travelling at 15 m/s sees a hazard 30 m ahead. Her reaction time is 0.7 s, during which the car keeps moving at 15 m/s. She then brakes with a constant deceleration of 5 m/s\(^2\).

1. Find the thinking distance. [1 mark]
2. Find the braking distance needed to stop. [3 marks]
3. Show that she cannot stop before reaching the hazard. [2 marks]
4. Find the speed at which she hits the hazard. [3 marks]

A stone is released from rest at the top of a bridge 45 m above the water. Take \(g=9.8\) m/s\(^2\).

1. Find the time to reach the water and the speed on impact. [4 marks]
2. A second stone is thrown downwards at 5 m/s from the same point. Show that it reaches the water when \(4.9t^2+5t-45=0\). [2 marks]
3. Find the time and impact speed for the second stone. [4 marks]

A lift accelerates uniformly from rest for 3 s at 1.2 m/s\(^2\), then travels at constant speed for 10 s, then decelerates uniformly to rest in 4 s.

1. Find the maximum speed. [1 mark]
2. Find the deceleration. [2 marks]
3. Find the total height climbed. [4 marks]
4. Find the average speed. [2 marks]

Runner \(A\) runs at a constant 7 m/s. Runner \(B\) starts 12 m behind \(A\) at the same instant from rest with a constant acceleration of 1.5 m/s\(^2\).

1. Write \(A\)\'s distance from \(B\)\'s starting point after \(t\) s. [1 mark]
2. Write \(B\)\'s distance after \(t\) s. [1 mark]
3. Show that \(B\) catches \(A\) when \(0.75t^2-7t-12=0\) and find \(t\). [4 marks]
4. How far has \(B\) run when she catches \(A\), and what is her speed then? [3 marks]

A lorry\'s acceleration is \(a=0.6-0.04t\) m/s\(^2\) for \(0\le t\le15\), where \(t\) is in seconds. Its initial speed is 2 m/s.

1. Show that \(v=2+0.6t-0.02t^2\). [2 marks]
2. Find the speed when \(t=15\) and explain why the acceleration is zero then. [3 marks]
3. Find the distance travelled in the 15 seconds. [4 marks]
4. Find the average speed over the 15 s. [1 mark]

A firework rocket is fired vertically upwards from the ground at 40 m/s and explodes at its highest point. Take \(g=9.8\) m/s\(^2\).

1. Find the height and time of the explosion. [4 marks]
2. Find the height after 3 s. [2 marks]
3. Suppose the rocket did not explode. For how long would it be above 70 m? [5 marks]

A car is travelling at 20 m/s on a straight road and accelerates uniformly at 0.8 m/s\(^2\) for a distance of 300 m.

1. Show that the time taken satisfies \(t^2+50t-750=0\). [3 marks]
2. Find the time. [2 marks]
3. Find the final speed. [2 marks]
4. Check the final speed using \(v^2=u^2+2as\). [2 marks]

A cyclist rides along a straight road in four stages.

1. Find the acceleration in stage 1 and the deceleration in stage 3. [3 marks]
2. Find the distance covered in each stage. [4 marks]
3. Find the total distance and the average speed. [3 marks]
4. The cyclist claims her average speed was 4 m/s for the whole ride. Is she right? [2 marks]

The velocity of a particle on a line is \(v=t^2-8t+12\) m/s for \(0\le t\le8\). At \(t=0\) the particle is at the origin.

1. Show that \(s=\dfrac{t^3}3-4t^2+12t\). [2 marks]
2. Find the displacement at \(t=2\), \(6\) and \(8\). [4 marks]
3. Find the total distance travelled in 8 seconds. [3 marks]
4. Find the acceleration at \(t=4\) and explain its sign in the table. [2 marks]

A car passes point \(A\) with speed \(u\) and accelerates uniformly at \(a\). It reaches point \(B\) after 4 s having travelled 60 m. It reaches point \(C\) 10 s after passing \(A\), having travelled 210 m from \(A\).

1. Write two equations in \(u\) and \(a\). [3 marks]
2. Solve them. [3 marks]
3. Find the speeds at \(B\) and \(C\). [2 marks]
4. Find the distance from \(B\) to \(C\) and the average speed between \(B\) and \(C\). [2 marks]

A ball is thrown vertically upwards at 12 m/s from the top of a cliff 25 m above the sea. It eventually falls into the sea. Take \(g=9.8\) m/s\(^2\) and upwards as positive.

1. Find the greatest height of the ball above the cliff top. [3 marks]
2. Show that the time to reach the sea satisfies \(4.9t^2-12t-25=0\) and solve it. [4 marks]
3. Find the speed when it hits the sea. [2 marks]
4. Find the total distance travelled by the ball. [3 marks]

A train travels 8 km between two stations in 300 s. It accelerates uniformly from rest for 40 s, then travels at constant speed \(V\) m/s, then decelerates uniformly at 0.5 m/s\(^2\) to rest.

1. Show that the time spent decelerating is \(2V\) seconds. [1 mark]
2. Show that the total distance is \(280V-V^2\) metres. [5 marks]
3. Find \(V\) to 1 decimal place. [3 marks]
4. Find the acceleration in the first stage and the time at constant speed. [3 marks]

Particle \(A\) is dropped from rest from a height of 80 m at the same moment as particle \(B\) is projected vertically upwards from the ground, directly below \(A\), with speed \(u\) m/s. Take \(g=9.8\).

1. Show that the particles meet after \(t=\dfrac{80}u\) seconds, whatever \(g\) is. [4 marks]
2. For \(u=40\) find the time of meeting and the height at which they meet. [3 marks]
3. Find \(u\) if they meet at a height of 20 m. [4 marks]