Kinematics Graphs (CIE A Level Maths: Mechanics)

A particle moves along a horizontal line starting at the point O. The displacement-time graph for the first 20 seconds of its motion is shown below. Displacement is measured in metres.

A TV camera runs along a horizontal track next to the pitch at a football match.

The pitch is 80 m long and the camera is initially positioned at the halfway line.
The camera moves from rest at a constant velocity and after 6 seconds has displacement 24 m.

It then moves back to the halfway line at a constant velocity, taking 12 seconds to do so.

For the next 6 seconds the camera moves with constant velocity . 

Sketch a displacement-time graph to illustrate the motion of the camera.

The diagram below shows the velocity-time graph for a model train travelling between two stations, starting at station A and finishing at station B. The graph indicates velocity in metres per second and time in seconds.

A cable car provides a scenic way to cross a river. It runs horizontally at a fixed height.
An individual cable car leaves one side of the river with a constant acceleration of for 5 seconds. It then moves at a constant speed until it has covered a distance of 45 m (at that speed). The cable car then decelerates uniformly until it comes to rest on the other side of the river 10 seconds later.

Sketch the velocity-time graph of the cable car’s journey across the river.

A ferry leaves port (point O) and travels directly towards its destination (point B). Its journey is shown on the displacement-time graph below. Displacement is measured in metres and time in seconds.

Use the graph to

The ferry company would like the journey to be completed within a maximum time of 25 minutes. Find, to three significant figures, the average velocity the ferry would need to travel at in order to achieve this.

A robot lawnmower is tested for accuracy by running it back and forth along a 20 m horizontal strip of grass. Starting in the middle of the strip of grass the lawnmower moves with a constant velocity of for 15 seconds. The velocity is then instantly changed to and this is maintained until the lawnmower has displacement 9 m from its starting position. The lawnmower is left in this position for 2 seconds before the final part of the test whereby it moves with a constant velocity of for 8 seconds.

Plot the displacement-time graph of the robot lawnmower on the axes below.

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, , at time  after being struck.

In drag racing two cars start from rest side-by-side and accelerate extremely rapidly in a straight horizontal line with the first to cross a finish line declared the winner.
Immediately after crossing the finish line the cars each release a parachute enabling them to decelerate to rest as quickly as possible.

The graph below shows the velocities, in metres per second, of two cars from the start of a race until they come to rest afterwards. 

Verify that both cars covered the distance of the drag racetrack, 300 m, before decelerating.

Without performing any calculations explain how you can tell that both cars had equal deceleration?

A person throws a pebble vertically upwards over the top of a cliff.

The displacement-time graph below describes the motion of the pebble until it lands in the sea below. At time t seconds the displacement of the pebble is .

Use the graph to

the time, other than when , at which the displacement of the pebble is zero,

A trainee is practising a take-off manoeuvre by piloting an aircraft along a straight horizontal runway. The velocity-time graph below describes the motion of the aircraft where velocity is measured in metres per second.

The aircraft accelerated at and the trainee pilot used 1.68 km of the runway to complete the practice manoeuvre.

 Show that

,

Hence find the values of V and T.

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.

A TV camera runs along a horizontal track next to the pitch at a football match.

The pitch is 100 m long and the camera is initially positioned at the halfway line.
The camera moves from rest at a constant velocity and after 5 seconds has displacement 35 m.

It remains in this location for 3 seconds before moving back to the halfway line at a constant speed of .

For the next 4 seconds the camera moves with constant velocity .

Sketch a displacement-time graph to illustrate the motion of the camera.

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.

A cable car provides a scenic way to cross a river. It runs horizontally at a fixed height.
An individual cable car leaves one side of the river with constant acceleration of for 5 seconds. It then moves at a constant speed until it has covered a further of 42 m. The cable car 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 cable car’s journey across the river.

A ferry leaves a port (point O) and travels in a straight line towards its destination (point C). Its journey is shown on the displacement time graph below where displacement is measured in metres and time in seconds.

Use the graph to

The ferry company would like to decrease the overall journey time from O to C by 5 minutes but can only increase the velocity of the ferry between points A and B.
Find the increase in velocity required in order for the journey time to be decreased appropriately.

A robot vacuum cleaner is tested for accuracy by running it back and forth along a 40 m horizontal strip of carpet. Starting 10m from one end of the carpet the vacuum cleaner moves with a constant velocity of for . The vacuum cleaner then changes direction and travels . For the final part of the test the vacuum cleaner travels with a constant velocity of for a further 4.8 m.

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, , at time  after being struck.

When the snooker ball comes to rest it has displacement 0.3 m. Find the value of T and the total distance travelled by the ball.

In drag racing two cars start from rest side-by-side and accelerate extremely rapidly in a straight horizontal line with the first to cross a finish line declared the winner.
Immediately after crossing the finish line the cars each release a parachute enabling them to decelerate to rest as quickly as possible.

The graph below shows the velocities, in metres per second, of two cars from the start of a race until they come to rest afterwards.

The drag race track (strip) is 400 m long. Find the value of V.

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

A person throws a pebble vertically upwards over the edge of a cliff. The displacement-time graph below describes the motion of the pebble until it lands in the sea below. At time t seconds the displacement of the pebble is .

The equation of the graph can be written in the form , where A and B are integers. Use the graph to determine the values of A and B.

Determine the maximum height above the sea the pebble reaches.

(i)        Find the average velocity of the pebble.

(ii)       Find the average speed of the pebble.

The velocity-time graph below describes the motion of a particle travelling in a horizontal line. Velocity is measured in metres per second.

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

Find the values of V and T.

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.

A TV camera runs along a horizontal track next to the pitch at a rugby match.

The pitch is long and the camera is initially positioned at the halfway line.
The camera moves from rest at a constant velocity and after 4 seconds has displacement . The camera then moves to a position 17 m over the halfway line at a constant speed of . It remains in this position for 6 seconds, then moves for the next 9 seconds with velocity .

Sketch a displacement-time graph to illustrate the motion of the camera.

Work out the average velocity of the camera during its motion.

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

A cable car provides a scenic way to cross a river. It runs horizontally at a fixed height.
An individual cable car leaves one side of the river with constant acceleration of for 3 seconds. It then moves at a constant speed until it has covered a further 24 m. The cable car then accelerates at a constant until its velocity is The cable car then comes to rest on the other side of the river 5 seconds later.

Sketch the velocity-time graph of the cable car’s journey across the river.

The diagram below shows the displacement-time graph for a ferry travelling in a straight line between two ports, indicated by points O and D on the graph. Displacement is measured in metres and time in seconds. (Note that points A, B and C do not represent ports.)

Work out the difference in velocity (in metres per second) between the fastest and slowest parts of the journey.

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. 

Show that

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.

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.