Equations of uniformly accelerated motion

Curling is a sport played on ice. Players slide stones made of granite along the ice. The aim of the sport is to slide a stone into a circular area called the target.

The diagram below shows two stones, A and B. Each stone has a mass of 18 kg.

Stone A is moving with a velocity of 2.1 m s^–1, as shown. B is initially stationary.

Show that the momentum of A is about 40 N s.

Stone A collides with stone B.

The diagram below shows the velocity of stone B after the collision. The point

Deduce whether stone A moves in the direction shown after the collision.

Stone A comes to rest 1.9 m from X. The deceleration of the stone on the ice

Deduce whether the collision was elastic.

In an experiment to determine the acceleration of free‑fall, g, a metal cylinder was dropped from rest down a glass tube. A light gate was positioned close to the outside of the glass tube, as shown.

The light gate was connected to a data logger that recorded the time t taken for the cylinder to pass the gate.

A student recorded the following data.

The student omitted t₁ when calculating the mean time. He justified this by stating that t₁ is an anomalous result. Another student stated that the correct mean time should use all the data in the table.

Explain who is correct.

Calculate the percentage uncertainty in the mean time t_mean.

The student used the mean time to calculate the velocity of the metal cylinder as it moved past the light gate.

State the extra measurement the student must have made to calculate this velocity.

The metal cylinder fell through a distance 0.75 m before reaching the light gate.

Calculate the velocity of the metal cylinder as it reached the light gate.

Explain why a light gate and data logger were used in this experiment rather than a stopwatch. You should refer to your answer to (b)‌(i).

An object was thrown so that it followed the path shown. Assume drag forces were negligible.

The object was thrown with an initial vertical component of velocity u. The time taken to reach maximum height is t.

Which of the following could not be used to determine the maximum vertical height s reached by the object?

The photograph shows a cyclist cycling at a constant velocity on horizontal ground.

The cyclist stops pedalling and comes to rest in a time of 5.2 s.

Sketch a graph to show how the cyclist’s velocity changes during this time.

Assume the deceleration is constant.

The cyclist travels 7.80 m while coming to rest.

Calculate the average resistive force on the cyclist and bicycle.

mass of cyclist and bicycle = 28.0 kg

In the game of golf a stationary ball is hit by a club. One of the aims of the game is to land the ball on a patch of ground called the green.

The graph shows how the force F exerted by the club on the ball varies with time t as the ball is hit.

State why the area under the graph represents impulse.

Show that the velocity of the ball is about 30 m s⁻¹ immediately after it is hit by the club.

mass of ball = 0.046 kg

The ball has a time of flight of 3.5 s before landing. The green is a vertical distance of 7.5 m above the point at which the ball was hit, as shown. The green is about seventy metres away from where the ball is hit.

Deduce whether, if air resistance is ignored, the ball could land on the green after a flight time of 3.5 s.

The photograph shows a model racing car set. The curved parts of the track are semicircular. The car makes electrical contact with the track using metal brushes underneath the car.

There is a maximum speed for the car to stay on the curved part of the track. Explain why the car will slip off the curved part of the track if the car exceeds the maximum speed.

The following measurements are made for a car starting at rest on a straight piece of track.

distance travelled = 1.2 m

time taken = 0.77 s

Show that the final velocity of the car is about 3 m s⁻¹.

Assume the acceleration is constant.

The final velocity calculated in (b)(i) is the maximum velocity before the car slips off the track.

Calculate the maximum horizontal force between the curved part of the track and the car.

mass of car = 0.050 kg

radius of curved part of track = 0.042 m

The cars are controlled separately and so can be raced, with one car on the inner lane and the other on the outer lane. The cars are identical. Each car is raced at its highest speed for that lane.

Explain why the outcome of the race is difficult to predict.

The photograph shows a model racing car set. The curved parts of the track are semicircular. The car makes electrical contact with the track using metal brushes underneath the car.

There is a maximum speed for the car to stay on the curved part of the track. Explain why the car will slip off the curved part of the track if the car exceeds the maximum speed.

The following measurements are made for a car starting at rest on a straight piece of track.

distance travelled = 1.2 m

time taken = 0.77 s

Show that the final velocity of the car is about 3 m s⁻¹.

Assume the acceleration is constant.

The final velocity calculated in (b)(i) is the maximum velocity before the car slips off the track.

Calculate the maximum horizontal force between the curved part of the track and the car.

mass of car = 0.050 kg

radius of curved part of track = 0.042 m

The cars are controlled separately and so can be raced, with one car on the inner lane and the other on the outer lane. The cars are identical. Each car is raced at its highest speed for that lane.

Explain why the outcome of the race is difficult to predict.

Scientists have been studying a type of jumping spider that can jump up to six times its body length.

The scientists photographed a spider at 0.02s intervals, during a jump. The picture is taken from the photograph and is shown actual size.

Deduce whether the images show that the motion in the x-direction is independent of the motion in the y-direction. You should take measurements using the cross marking the centre of gravity of the spider.

Show that the initial velocity of the spider at the start of the jump is about 1 m s⁻¹. You should take measurements using the cross marking the centre of gravity of the spider.

The spider achieves this jump by extending its two back legs by 3.0 mm.

Calculate the average force the spider exerts in each leg to achieve the jump.

mass of spider = 150 mg

Just as the spider starts the jump, it fixes a silk thread to the platform. It is thought that the thread acts as a safety line in case the spider falls.

A student makes the comment:

ʻIf the silk thread can withstand a tension equal to the weight of the spider then this safety system should work.ʼ

Deduce whether this statement is correct.

Scientists have been studying a type of jumping spider that can jump up to six times its body length.

The scientists photographed a spider at 0.02s intervals, during a jump. The picture is taken from the photograph and is shown actual size.

Deduce whether the images show that the motion in the x-direction is independent of the motion in the y-direction. You should take measurements using the cross marking the centre of gravity of the spider.

Show that the initial velocity of the spider at the start of the jump is about 1 m s⁻¹. You should take measurements using the cross marking the centre of gravity of the spider.

The spider achieves this jump by extending its two back legs by 3.0 mm.

Calculate the average force the spider exerts in each leg to achieve the jump.

mass of spider = 150 mg

Just as the spider starts the jump, it fixes a silk thread to the platform. It is thought that the thread acts as a safety line in case the spider falls.

A student makes the comment:

ʻIf the silk thread can withstand a tension equal to the weight of the spider then this safety system should work.ʼ

Deduce whether this statement is correct.

initial final

The student read that the frictional force between an object and a surface is directly

In the sport of curling, two teams of ‘curlers’ take turns sliding polished granite stones across an ice surface towards a circular target marked on the ice.

A stone of mass 19.6 kg is accelerated uniformly for 1.25 s before being released by a curler. The stone then decelerates uniformly to rest, travelling 32.5 m in a time of 17.5 s.

Calculate the average useful power developed by the curler in accelerating the stone.

*(b) Stone B is stationary. Stone A travels towards the target and makes a direct hit on stone B as shown. Both stones have mass m.

The collision is elastic. Just before the collision stone A has a velocity v. After the collision stone B moves off with velocity v.

Discuss how the relevant conservation laws apply to this collision.

While a stone is moving towards the target, the curlers vigorously sweep the ice directly in front of the stone.

Explain why this may make the stone travel further.

The Shanghai Maglev Train is the first commercially operated high-speed magnetic levitation train in the world, connecting the airport and central Shanghai.

The total distance travelled is 29.9 km and the total journey time is 440 s. The train starts from rest and reaches a speed of 97 m s⁻¹ in 120 s.

Calculate the average acceleration of the train during the first 120 s.

Calculate the average speed of the train for the period following the 120 s acceleration.

Electromagnetic forces enable the train to levitate above a steel rail.

Explain why magnetic levitation is an advantage for a high-speed transport system.

*(c) A linear induction motor provides the force to accelerate the train forwards. A current flows in sequence through coils of wire mounted in the track. The train is dragged along as the magnetic field progresses along the coils of wire in the track. This is similar to moving a permanent magnetic field away from a conductor.

A teacher demonstrates this effect by quickly removing one end of a bar magnet from a suspended aluminum ring.

When the magnet is removed from the ring, the ring moves in the same direction as the magnet.

Explain, using the laws of electromagnetic induction, why the ring moves in the direction of the magnet.

The shot put is an Olympic field event. The distance for the women’s world record shot put is in excess of twenty two metres.

The shot is a metal ball, which is thrown from a standing position so that it lands on the ground a horizontal distance R away from the thrower.

A shot is thrown from a height h above the ground with an initial velocity of 14.2 m s^–1 at an angle of 37.0° to the horizontal. The time it takes for the shot to reach the ground is 1.98 s.

Show that the vertical component of the initial velocity is about 8.6 m s^–1.

Calculate the height h above the ground from which the shot was thrown.

Calculate the horizontal distance R for this throw.

A student investigates the motion of a friction‑free trolley down a ramp. On the top of the trolley there is a metal loop which makes contact with a length of thin resistance wire, AB, fixed above the ramp. The resistance wire has a uniform diameter.

The trolley accelerates down the ramp and the metal loop stays in contact with the wire along the full length of the ramp.

The student uses a protractor to measure the angle θ between the ramp and the horizontal

and records a value of 4° with an uncertainty of ±1°.

The two ends of the wire are connected to a 1.5 V cell. A data logger, set to measure potential difference, is connected to the metal loop and to the negative terminal of the cell.

Explain how the potential difference recorded by the data logger will vary as the loop moves along the length of the wire AB.

The graph shows the data obtained from the data logger.

Determine the velocity of the trolley at 1.5 s.

1.5 V represents a distance of 2.00 m.

The student calculated the velocity of the trolley at 2.0 s to be 1.5 m s^–1.

By considering the acceleration of the trolley, determine whether the student’s measurement of θ was within the uncertainty quoted.