Equations of uniformly accelerated motion

In 2021 the world land speed record was 1230 km h⁻¹. This was the average speed achieved by a jet-powered car in two runs. Each run was measured over a distance of 1.61 km.

The average speed for one of these runs was 343 m s⁻¹.

Calculate, in s, the time taken for the car to complete the other run.

Engineers are designing a new jet-powered car to break this record.

Figure 1 shows the variation of speed with distance for the car, as predicted by the engineers.

The car reaches its maximum acceleration when it is 5600 m from the start. At this point the mass of the car is 6.50 × 10³ kg.

Determine the kinetic energy of the car at its maximum acceleration.

At any point on the graph in Figure 1, the acceleration is given by:

acceleration = speed × gradient of line

When the car is at its maximum acceleration, the power input to the jet engines is 640 MW.

Calculate the percentage of the input power used to accelerate the car at its maximum acceleration.

Scientists recommend that the average deceleration of the driver of the car should be less than 3 g.

Deduce whether the average deceleration is less than 3 g.

Which combination of an object’s speed and journey time gives a distance travelled

Two circular discs made of card rotate at constant speed on a common axle.

The discs are 2.00 m apart.

An air-gun pellet is fired parallel to the axle. The pellet makes holes in the discs. The holes are separated by an angle of 45°. The speed of the pellet between the discs is 300 m s⁻¹.

How many revolutions does each disc complete in one second?

A stroboscope emits bright flashes of white light. The duration of each flash and the frequency of the flashes can be varied.

Table 1 shows information about the stroboscope.

The duration of each flash is T₁. The time from the start of a flash to the start of the next flash is T₂.

The duty cycle of a stroboscope is defined as .

What is the maximum duty cycle of the stroboscope?

Tick () one box.

Figure 1 shows images produced in an experiment in which a bouncing ball is illuminated by a stroboscope. The stroboscope flashes at a constant frequency.

Suggest why T₁ must be very short for this experiment.

Figure 2 shows the first six images starting with n = 0, where n is the image number.

The images are used to determine: H, the vertical distance from the bottom of the ball to the floor when n = 0 h, the vertical distance from the bottom of the ball to the floor for each non-zero value of n.

The n = N image is produced at the instant that the ball hits the floor for the first time. For n between 0 and N it can be shown that

where u₀ is the vertical velocity of the ball when n = 0 g is the acceleration due to gravity f is the frequency of the flashes.

In order to find g, a graph is plotted with values of on the y

Suggest what is plotted on the x-axis. Go on to explain how g is determined from this graph.

The following data are recorded.

H = 1550 mm f = 31.0 Hz

The graphical analysis of data from Figure 1 gives g as 9.79 m s⁻².

Determine u₀.

Figure 3 shows positions of the bottom of the ball for n = 40 to n = 66 In this range of positions, the ball makes contact with the floor for the second and third times. Values of h, the vertical distance from the bottom of the ball to the floor, are plotted on the y-axis. Values of s, the horizontal displacement from a point on the floor below the centre of the n = 0 image, are plotted on the x-axis.

Determine, in mm s⁻¹, the horizontal velocity of the ball between the second and third contacts of the ball with the floor.

Determine the time between the second and third contacts. Annotate Figure 3 to show your method.

A ball is thrown vertically upwards and returns to its original position 2.4 s later. The effect of air resistance is negligible.

What is the total distance travelled by the ball?

A tennis ball has a mass of 58 g. The ball is dropped from rest from a height of 1.8 m above the ground and falls vertically. The ball rebounds vertically to a height of 1.1 m. The effect of air resistance is negligible.

What is the change in momentum of the ball during its collision with the ground?

Figure 8 shows the H-shaped posts used in a game of rugby.

Figure 9 shows the path of a ball that is kicked and just passes over the crossbar. The initial velocity of the ball is 20.0 m s⁻¹ at an angle of 40.0° to the ground.

You should consider air resistance to be negligible and treat the ball as a simple projectile.

The top of the crossbar is 3.00 m above the horizontal ground.

Show that the minimum speed of the ball in flight is about 15 m s⁻¹.

Explain your answer.

The ball just passes over the crossbar at a time t after it is kicked.

Show that t must satisfy the following equation:

There are two solutions to the equation

Discuss which of the two solutions is the time taken for the ball to pass over the crossbar from when it is kicked.

In your answer you should

• state the value for t given by each solution

• explain the physical significance of the other solution.

Another attempt is made to kick the ball over the crossbar. The initial velocity of the ball is the same as in the first attempt. This kick is made from a horizontal distance of 38 m from the posts.

Deduce whether the ball can pass over the crossbar.

Figure 10 shows the variations with time of the vertical velocity of a ball with and without air resistance.

Discuss the features of the motion of the ball shown by the two graphs.

In your answer you should refer to

• the gradients of the graphs

• the area between each line and the time axis.

A pair of cameras is used on a motorway to help determine the average speed of vehicles travelling between the two cameras.

Figure 5 shows the speed–time graph for a car moving between the two cameras.

The speed limit for the motorway between the two cameras is 22 m s⁻¹.

Determine whether the average speed of the car exceeded this speed limit.

Markings called chevrons are used on motorways.

The chevron separation is designed to give a driver time to respond to any change in speed of the car in front. The driver is advised to keep a minimum distance d behind the car in front, as shown in Figure 6.

Government research suggests that the typical time for a driver to respond is between 1.6 s and 2.0 s.

Suggest a value for d where the speed limit is 31 m s⁻¹.

The chevron separation is based on the response time, not on the time taken for a car to stop.

The brakes of a car are applied when its speed is 31 m s⁻¹ and the car comes to rest. The total mass of the car is 1200 kg.

The average braking force acting on the car is 6.8 kN.

Calculate the time taken for the braking force to stop the car and the distance travelled by the car in this time.

Suggest why the chevron separation on motorways does not take into account the distance travelled as a car comes to rest after the brakes are applied.

At bends on motorways the road is sloped so that a car is less likely to slide out of its lane when travelling at a high speed.

Figure 7 shows a car of mass 1200 kg travelling around a curve of radius 200 m. The motorway is sloped at an angle of 5.0°.

Figure 8 shows the weight W and reaction force N acting on the car. The advisory speed for the bend is chosen so that the friction force down the slope is zero.

Suggest an appropriate advisory speed for this section of the motorway.

Small water drops leave a tap with zero velocity at intervals of 0.20 s. They then fall freely 0.80 m to reach a horizontal surface.

How far has a drop fallen when the previous drop hits the surface?

Two cars P and Q leave from the same point and travel in the same direction. Q leaves at time t = 0 and P leaves one second later. The figure shows the velocity – time graph for P and Q.

What is the distance between Q and P when t = 8 s ?

Horizontal escape lanes made of loose gravel have been constructed at the side of some roads on steep hills so that vehicles can stop safely when their brakes fail.

Figure 1 shows an engineer’s prediction of how the speed of an unpowered vehicle of mass 1.8 × 10⁴ kg will vary with time as the vehicle comes to rest in an escape lane.

Determine the force decelerating the vehicle 2.0 s after entering the escape lane.

Deduce whether a lane of length 85 m is long enough to stop the vehicle, assuming that the engineer’s graph is correct.

Discuss the energy transfers that take place when a vehicle is decelerated in an escape lane.

An alternative to an escape lane containing gravel is an escape lane that consists of a ramp. An escape ramp is a straight road with a concrete surface that has a constant upward gradient.

One escape ramp makes an angle of 25° to the horizontal and is 85 m long.

Deduce whether this escape ramp is sufficient to stop the vehicle.

Assume that any frictional forces and air resistance that decelerate the vehicle are negligible.

Discuss whether an escape lane containing gravel or an escape ramp would provide the safer experience for the driver of the vehicle as it comes to rest.

Figure 8

The model consists of two identical trolleys of mass M on a ramp which is at 35° to the horizontal. The trolleys are connected by a wire that passes around a pulley of negligible mass at the top of the ramp.

Two concrete blocks each of mass m are loaded onto trolley A at the top of the ramp. The trolley is released and accelerates to the bottom of the ramp where it is stopped by a flexible buffer. The blocks are unloaded from trolley A and two blocks are loaded onto trolley B that is now at the top of the ramp. The trolleys are released and the process is repeated.

Figure 9 shows the side view of trolley A when it is moving down the ramp.

The tension in the wire when the trolleys are moving is T.

Draw and label arrows on Figure 9 to represent the magnitudes and directions of any forces and components of forces that act on trolley A parallel to the ramp as it travels down the ramp.

Assume that no friction acts at the axle of the pulley or at the axles of the trolleys and that air resistance is negligible.

Show that the acceleration a of trolley B along the ramp is given by

Compare the momentum of loaded trolley A as it moves downwards with the momentum of loaded trolley B.

In practice, for safety reasons there is a friction brake in the pulley that provides a resistive force to reduce the acceleration to 25% of the maximum possible acceleration.

The distance travelled for each journey down the ramp is 9.0 m.

The following data apply to the arrangement.

Mass of a trolley M = 95 kg Mass of a concrete block m = 30 kg

Calculate the time taken for a loaded trolley to travel down the ramp.

It takes 12 s to remove the blocks from the lower trolley and reload the upper trolley.

Calculate the number of blocks that can be transferred to the lower level in 30 minutes.