Graphs in Kinematics (Edexcel A Level Maths: Mechanics): Exam Questions

A tram pulls away from stop A, accelerating uniformly from rest to 20 ms⁻¹ in 10 seconds.
It then travels at this constant speed for another 30 s.
During the final 20 s of the trip it decelerates uniformly to rest at stop B.

The velocity–time graph for the journey is shown in Figure 1.

Find the distance between stop A and stop B.

Determine the magnitude of the tram’s deceleration during the last 20 s of its motion.

A snooker ball is struck so that it travels in a straight line along the length of a table, rebounds from a cushion and then rolls back towards its starting point.

Figure 1 shows the velocity–time graph for the ball.

Velocity is measured in metres per second and time is measured in seconds.

Calculate, in metres per second, the reduction in the ball’s speed as a result of the impact with the cushion.

Calculate the acceleration of the snooker ball before it hits the cushion.

Describe the acceleration, the velocity, and the speed of the ball between and .

You do not need to calculate any numerical values.

Find the final displacement of the ball from its starting position.

Two drag-racing cars, and , start together from rest.

They accelerate in a straight horizontal line to their respective top speeds, then maintain that speed until they cross the finish line.

The instant each car reaches the finish line it releases a parachute and decelerates uniformly to rest.

Figure 1 shows the velocity–time graphs for the two cars from the start of the race until each comes to rest.
Velocity is measured in metres per second and time in seconds.

For the car which won the race, calculate the magnitude of its initial acceleration.

Show that both cars covered the distance of the race track, 300 m, before decelerating.

Without carrying out any further calculations, explain how, by using the velocity–time graph, you can deduce that cars and experienced the same deceleration after they crossed the finish line.

Figure 1 shows the velocity–time graph for an aircraft as it carries out a practice manoeuvre along a straight, horizontal runway.
Velocity is measured in metres per second and time in seconds.

• The aircraft starts from rest at

• It accelerates uniformly with acceleration 5 ms⁻² for the first seconds until its speed is  ms⁻¹

• It then continues at the constant speed  ms⁻¹ for a further 10 seconds

• Finally it decelerates uniformly to rest, coming to rest at time seconds after starting its motion

During the whole manoeuvre the aircraft travels 1.68 km along the runway.

Find the values of and .

A television camera runs on a straight, horizontal rail that lies parallel to the long side of a football pitch, where .

A fixed origin is taken at the halfway line, with the positive direction from towards .

The motion of the camera is described below.

• At time the camera is at rest at .

• It then moves with a constant velocity so that, at , its displacement from is

• The camera remains at this point for a further

• It then travels back towards with a constant speed of until it reaches

• Immediately after passing through it continues along the rail with a constant velocity of for

Determine the displacement of the camera from at the end of the motion described above and state the total time taken to reach this position.

Hence determine the total distance travelled by the camera from until it reaches the final displacement found in part (a).

The diagram below shows the velocity-time graph for a train travelling between two stations, starting at station P and finishing at station Q. The graph indicates velocity in kilometres per hour and time in minutes.

Find the distance between station P and station Q.

Find the deceleration of the train in the last 20 minutes.

A particle is moving along a straight horizontal line.

The particle starts at rest and then accelerates at for 5 seconds. It then moves at a constant speed until it has covered an additional 42 m.

The particle then accelerates at a constant until its velocity is . It remains at this velocity for 5 seconds before taking a further 6 seconds to come to rest at a uniform rate.

Sketch the velocity-time graph of the particle's motion.

A particle moves along a horizontal path, its motion, in terms of displacement from point O, is shown for the first 20 seconds on the displacement-time graph below. Displacement is measured in metres and time is measured in seconds.

Find the distance travelled by the particle in the first 10 seconds.

Work out the particle’s average speed in the first 20 seconds.

A cable-car carries passengers horizontally across a river at a constant height above the water.

The motion of one journey is as follows.

• From rest, the cable-car accelerates uniformly at 2 ms^-2 for 3 seconds

• It then continues at the speed reached until it has travelled a further 24 m

• The cable-car next decelerates uniformly with acceleration -1 ms^-2 until its velocity is -2 ms^-1

• Finally, the cable-car decreases its speed to come to rest 5 s later on the opposite bank of the river

Sketch a velocity–time graph for the complete motion of the cable-car described.

Find the total distance the cable car travels in the 20 seconds.

A snooker ball is struck so that it moves in a straight line along the length of a perfectly horizontal table.

The velocity of the ball at time after it is struck is shown in Figure 1.

• The ball first travels away from the cushion with constant deceleration, its velocity decreasing linearly from at to at

• The ball then strikes the cushion at  and rebounds along the same line, its velocity instantaneously changing to

• Finally the ball moves back towards the original position with constant acceleration, coming to rest at

Using information from Figure 1, show that the magnitude of the acceleration of the ball after it hits the cushion is

When the snooker ball comes to rest it has displacement 0.3 m.

Find the value of and the total distance, , travelled by the ball.

Two drag-racing cars, and , start together from rest.

The instant each car reaches the finish line it releases a parachute and decelerates uniformly to rest.

Figure 1 shows the velocity–time graphs for the two cars from the start of the race until each comes to rest.
Velocity is measured in metres per second and time in seconds.

One of the cars was still accelerating when it crossed the finish line. Work out the magnitude of the acceleration for this car as it approaches the finish line.

The drag race track (strip) is 400 m long. Find the value of , which is indicated on Figure 1.

Hence show that half the distance covered by car was while decelerating.

A person throws an object vertically upwards from the top of a platform. Figure 1 shows a displacement-time graph for the motion of the object until it lands on the ground below the platform.

At time seconds the displacement of the object is  metres.

Given that the equation of the graph can be written as

where and  are integers, use Figure 1 to determine the values of and .

Determine the maximum height above the platform that the object reaches.

Find

(i) the average velocity of the pebble,

(ii) the average speed of the pebble.

Figure 1 shows a velocity-time graph for the motion of a particle travelling in a straight horizontal line. Velocity is measured in metres per second and time is measured in seconds.

The particle has a final displacement of −27 m and its greatest magnitude of acceleration throughout the motion was .

Find the values of and , which are labelled in Figure 1.

A train travels along a straight horizontal track from station to station .

In a model of the motion of the train, at time the train starts from rest at , and moves with constant acceleration until it reaches its maximum speed of 25 ms⁻¹.

The train then travels at this constant speed of 25 ms⁻¹ before finally moving with constant deceleration until it comes to rest at .

The time spent decelerating is four times the time spent accelerating.

The journey from to takes 700 s.

Using the model, sketch a speed-time graph for the motion of the train between the two stations and Q.

The distance between the two stations is 15 km.

Using the model, show that the time spent accelerating by the train is 40 s.

Using the model, find the acceleration, in ms⁻², of the train.

Using the model, find the speed of the train 572 s after leaving .

State one limitation of the model which could affect your answers to parts (b) and (c).

Two children, Pat () and Sam (), run a race along a straight horizontal track.

Both children start from rest at the same time and cross the finish line at the same time.

In a model of the motion:

Pat accelerates at a constant rate from rest for 5s until reaching a speed of 4 ms^-1 and then maintains a constant speed of 4 ms^–1 until crossing the finish line.

Sam accelerates at a constant rate of 1 ms^–2 from rest until reaching a speed of ms^–1 and then maintains a constant speed of ms^–1 until crossing the finish line.

Both children take 27.5 s to complete the race.

The velocity-time graphs shown in Figure 1 describe the model of the motion of each child from the instant they start to the instant they cross the finish line together.

Using the model, explain why the areas under the two graphs are equal.

Using the model, find the acceleration of Pat during the first 5 seconds.

Using the model, find, in metres, the length of the race.

Using the model, find the value of , giving your answer to 3 significant figures.

Figure 1 shows the displacement–time graph for a ferry that travels in a straight line between two ports, and .
The ferry leaves port , passes points , and and finally comes to rest at port D.
Displacement is measured in metres and time in seconds.

Note that , and are not ports.

The motion consists of four straight-line sections.

The speed of the ferry is constant along each straight-line section.

Calculate, to the nearest metre per second, the difference between the greatest and the least of these constant speeds.

The ferry company would like to decrease the overall time the journey takes, but can only increase the velocity of the ferry whilst it is further than 5 km from a port. Given that the maximum speed of a ferry is . Find, to the nearest second, the time by which the ferry company is able to reduce the journey time by.

A robot designed to walk like a human being is tested for accuracy by being programmed to walk back and forth in a horizontal line.

In a three-part test the robot walks with average velocities of , in that order.

The first two parts of the robot’s walk produce an average velocity of and the last part of the walk lasts 10 seconds. The final displacement of the robot is 4 m.

Plot the displacement-time graph of the robot on the axes below, labelling the coordinates of the points where the robot’s velocity changes and the coordinates of its final position.

A snooker ball is struck such that it travels in a straight line up and down a snooker table. The graph below shows the velocity of the ball, v m s⁻¹, at time .

Write down the times at which the snooker ball hits a cushion and hence, find the length of the snooker table.

Show that the snooker ball comes to rest in exactly the same place it was struck from.

In a handicapped drag race two cars start side-by-side from rest but one driver is given a 2 second head-start. Drag cars accelerate extremely rapidly, moving in a straight horizontal line with the first car to cross a finish line declared the winner. Immediately after crossing the finish line each car deploys a parachute enabling it to come to rest as quickly as possible. The graph below shows the velocities, in metres per second, of two cars from the start of a handicapped race until they come to rest afterwards. 

(i) Show that

(ii) Given that the length of a drag race is 400 m, find the values of V₁ and V₂.

Find the time at which, before they cross the finish line, the two cars have the same velocity.

Two pebbles are thrown over the edge of the cliff at the same time. Pebble P is thrown such that it only travels vertically downwards. Pebble Q is thrown vertically upwards. The graph below shows the displacement, in metres, of both pebbles at time t seconds, until they land in the sea.

(i) Label the curves on the graph above with Pebble P  and Pebble Q as appropriate.

(ii) Write down the height of the cliff.

(iii) One of the pebbles spends half of its journey time to the sea with positive displacement. Write down the time it takes both pebbles to land in the sea.

Without performing any calculations explain how you know that both pebbles have the same average velocity at the time they land in the sea.

The displacement of the pebble thrown upward can be described by the equation s = At(B − t), where A and B are integers. Find the maximum height above the sea this pebble reaches and how long after being thrown it takes to reach this height.

The estimated range of a mobility scooter (how far it can travel on a single battery charge) is calculated by running the scooter on a machine that would be the equivalent of driving it along a straight horizontal road.

A scooter is set to move from rest with constant acceleration for 4 seconds. The scooter then maintains a constant velocity for the next 16 minutes. The velocity of the scooter is reduced by a quarter which takes 5 seconds. The scooter runs at this velocity for a further 13.5 minutes. Then, as the battery runs out, the scooter takes 25 seconds to come to rest.

Show that the range of this scooter should be estimated as 12.6 km and that the battery lasts for approximately half an hour on a single charge.