Graphs in Kinematics (AQA 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 $v$is measured in metres per second and time $t$ 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 $t=0.8$ and $t=2.3$.

You do not need to calculate any numerical values.

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

Two drag-racing cars, $A$ and $B$, 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 $v$ is measured in metres per second and time $t$ 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 $A$ and $B$ 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 $t=0$

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

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

• Finally it decelerates uniformly to rest, coming to rest at time $2T$ seconds after starting its motion

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

Find the values of $V$ and $T$.

A television camera runs on a straight, horizontal rail that lies parallel to the long side $AB$ of a football pitch, where $AB=100\mathrm{m}$.

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

The motion of the camera is described below.

• At time $t=0$ the camera is at rest at $O$.

• It then moves with a constant velocity so that, at $t=5\mathrm{s}$, its displacement from $O$ is $35\mathrm{m}$

• The camera remains at this point for a further $3\mathrm{s}$

• It then travels back towards $O$ with a constant speed of $10{\mathrm{ms}}^{-1}$ until it reaches $O$

• Immediately after passing through $O$ it continues along the rail with a constant velocity of $-2.5{\mathrm{ms}}^{-1}$ for $4\mathrm{s}$

Determine the displacement of the camera from $O$ 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 $t=0$ 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 $1.2\mathrm{m}{\mathrm{s}}^{-2}$ 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 $-0.5\mathrm{m}{\mathrm{s}}^2$ until its velocity is $1.5\mathrm{m}{\mathrm{s}}^{-1}$. 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 $v{\mathrm{ms}}^{-1}$of the ball at time $t\mathrm{s}$ 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 $5{\mathrm{ms}}^{-1}$ at $t=0$ to $3{\mathrm{ms}}^{-1}$ at $t=0.6\mathrm{s}$

• The ball then strikes the cushion at $t=0.6\mathrm{s}$ and rebounds along the same line, its velocity instantaneously changing to $-1.5{\mathrm{ms}}^{-1}$

• Finally the ball moves back towards the original position with constant acceleration, coming to rest at $t=T\mathrm{s}$

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

$\frac{15}{10T-6}{\mathrm{ms}}^{-2}$

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

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

Two drag-racing cars, $A$ and $B$, 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 $v$ is measured in metres per second and time $t$ 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 $V$, which is indicated on Figure 1.

Hence show that half the distance covered by car $B$ 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 $t$ seconds the displacement of the object is $s$ metres.

Given that the equation of the graph can be written as

$s=At(B-t)$

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

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 $1.5\mathrm{m}{\mathrm{s}}^{-2}$.

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

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

Note that $A$, $B$ and $C$ 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 operator wishes to reduce the total journey time.
To achieve this, the vessel will travel at the fleet’s maximum allowed speed of 45 ms⁻¹ when it is more than 5 km from either port.

Assuming the motion is otherwise unchanged, determine by how much the total journey time can be reduced. Give your answer to the nearest second.

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 similar to driving it along a straight horizontal road.

A scooter is set to move with the following pattern:

• Starting from rest, it moves with constant acceleration $2{\mathrm{ms}}^{-2}$ 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 decelerates uniformly to rest, taking 25 seconds to come to rest

Show that

• The range of this scooter should be estimated as 12.6 km

• The battery lasts for approximately half an hour on a single charge

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 $-0.5\mathrm{m}{\mathrm{s}}^{-1},1.5\mathrm{m}{\mathrm{s}}^{-1}\mathrm{and}-1\mathrm{m}{\mathrm{s}}^{-1}$, in that order.

The first two parts of the robot’s walk produce an average velocity of $0.7\mathrm{m}{\mathrm{s}}^{-1}$ 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.

In a staggered drag race two cars start side-by-side from rest but one driver is given a 2 second head-start.

The cars accelerate extremely rapidly, moving in a straight horizontal line with the first car to cross the finish line declared the winner.

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 $v$ is measured in metres per second and time $t$ in seconds.

Given that the length of a drag race is 400 m, find the values of $V_1$ and $V_2$ which are labelled in Figure 1.

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