4.3 Equations of Motion

Figure 1 below shows a skateboarder descending a ramp. 

Figure 1

The skateboarder starts from rest at the top of the ramp at A and leaves the ramp at B horizontally. In going from A to B the skateboarder’s centre of gravity descends a vertical height of 1.8 m. After leaving the ramp at B, the skateboarder lands on the ground at C, 0.62 m away. 

Calculate the magnitude and direction of the horizontal resultant velocity immediately before impact at C. Take the downwards direction as positive. 

In this question, assume that air resistance is negligible.

The skateboarder now leaves a different ramp inclined at 30º and 1.5 m high and lands on another ramp 55 m away which has a height of 1 m, as shown in Figure 2. 

Figure 2

Calculate the speed which the skateboard needs in order to reach the other ramp.

An archer of height 1.8 m shoots an arrow from a bow. 

Each arrow has a mass of 70 g and the bow string is displaced by a maximum of 0.65 m before the arrow is released. The archer exerts a maximum force of 550 N and the arrow is aimed at an angle of 60º to the horizontal. 

Calculate the total time that the arrow is in flight if it lands on the ground. 

Assume there is no friction between the arrow and bow during release.

The archer, inspired by the Hunger Games, decides to practice hitting a moving target with their arrow. They see a tree 10 m away with a loosely hung apple at a height of 5.8 m from the ground. Once the arrow leaves the archer’s bow at a velocity of 70 m s^–1, their friend shakes the tree branches and the target apple immediately falls vertically downwards in freefall, as shown in Figure 2. 

Figure 2

The archer is debating whether they should aim above the apple (path A), at the apple (path B) or directly below the apple (path C) to hit it. 

Show, with a calculation, that the archer should aim directly at the apple with path B.

A stone thrown from the top of a building is given an initial velocity of 18.0 m s^–1 straight upwards. The building is 60.3 m high and the stone just misses the roof on its way down. 

Calculate the time taken for the stone to 

For this question, take the upwards direction as positive and time t = 0 s is when the stone is first thrown upwards. Assume that air resistance is negligible throughout.

Calculate the velocity of the stone at the instant it: 

For this question, take the upwards direction as positive and time t = 0 s is when the stone is first thrown upwards.      

Draw a velocity time graph of the motion of the stone, clearly marking the points calculated from part (a) and (b).

When the stone hits the bottom of the building, it bounces upwards again at a velocity of 28 m s^–1. It then continues to bounce, losing a quarter of its kinetic energy each time.

Figure 1

Sketch the motion of the stone on the graph in Figure 1 for two bounces. Clearly label any values on the velocity axis.     

A bus driver drives with a constant acceleration of 6.0 m s^–2. After 3.0 seconds, they slow down to stop at a set of traffic lights at 5.0 m s^–2. 

Given that the bus initially travelled at 4.0 m s^–1 and that the bus fully stops at the traffic lights, calculate the distance between the traffic lights and the point where the bus began to accelerate. 

In this question, assume that air resistance is negligible.

Figure 1

Draw the velocity­–time graph for the motion of the bus on Figure 1.

Figure 2

Sketch the displacement–time and acceleration­–time graphs in Figure 2 for the bus. Label the axes appropriately and the graphs do not have to be to scale.

A hot–air balloonist drops an apple over the side at a height of 100 m whilst it is travelling upwards. The speed of the balloon is 3.5 m s^–1 at the moment the apple is released. 

Calculate: 

   (i)    the time taken for the apple to fall to the ground. 

   (ii)    the impact velocity of the apple on the ground. 

Take the upwards direction as positive and assume that air resistance is negligible.

Now assume that air resistance is non–negligible. The apple reaches a velocity of 25 m s^–1 after 3.9 s where it then has zero resultant force for the final 40 m. 

Sketch a velocity­–time graph of the motion of the apple. Label any relevant times and velocities of the apple.

The apple is still dropped from 100 m, and air resistance is still taken into consideration. 

Explain the motion of the apple up to and including when it has zero resultant force.

The hot–air balloonist now has two apples, Apple A and Apple B. Apple A is much lighter than apple B and they are both released from a stationary hot–air balloon at the same distance above the ground. 

Explain which apple should be thrown from the hot–air balloon second, in order for them both to reach the ground at the same time.

A digital camera was used to obtain a sequence of images of a tennis ball being struck by a tennis racket. The camera was set to take an image every 6.0 ms. The successive positions of the racket and ball are shown in Figure 1 below.

Figure 1

The ball has a horizontal velocity of zero at A and reaches a constant horizontal velocity at D as it leaves the racket. The ball travels a horizontal distance of 96 cm between D and G. 

Calculate the horizontal acceleration of the ball between A and D.

At D, the ball was projected horizontally from a height of 3.4 m above level ground. 

Show that the ball would fall to the ground in about 0.8 s. 

Assume that only gravity acts on the ball as it falls.

Calculate the horizontal distance that the ball will travel after it leaves the racket before hitting the ground. 

Assume that only gravity acts on the ball as it falls.

Explain why, in practice, the ball will not travel this far before hitting the ground.

Figure 1 shows the variation of velocity v with time t for a Formula 1 car during a test drive along a straight, horizontal track. 

Figure 1

State and explain which section of the graph shows that the car’s acceleration was uniform.

Determine the distance travelled by the car during the first 5.0 s.

Show that the instantaneous acceleration is about 16 m s⁻² when t is 5.0 s.

Figure 2 shows the aerofoil that is fitted to a Formula 1 car to increase its speed around corners. 

Figure 2

However, the aerofoil exerts an unwanted drag force on the car when it is travelling in a straight line so a Drag Reduction System (DRS) is fitted. This system enables the driver to change the angle of the aerofoil to reduce the drag. 

The graph in Figure 1 is for a test drive along a straight, horizontal track. Under the conditions for this test drive, the DRS was not in use and the engine produced a constant driving force. 

Explain why the velocity varies in the way shown in the graph. 

Go on to explain how the graph will be different when the DRS is in use and the driving force is the same. 

The quality of written communication will be assessed in your answer.

Figure 1 shows a golfer hitting a ball from the top of a cliff. 

Figure 1

The ball follows the path shown. The ball is hit with an initial velocity of 46 m s⁻¹ at an angle of 34° above the horizontal, as shown. Assume that there is no air resistance. 

Calculate the initial vertical component of velocity of the ball.

At point Y the ball is level with its initial position. 

Show that the time taken to reach Y is about 5 s.

The total time of flight of the ball is 10.0 s. 

Show on Figure 2 how v, the vertical component of the velocity, changes throughout the whole 10.0 s. 

Figure 2

Calculate the height h of the cliff.       

In practice, the air resistance affects the path of the ball. 

Draw on Figure 1the path the ball takes when air resistance is taken into account.

The graph shows how the vertical speed of a parachutist changes with time during the first 20 s of his jump. To avoid air turbulence caused by the aircraft, he waits a short time after jumping before pulling the cord to release his parachute. 

Figure 1

Regions A, B and C of the graph show the speed before the parachute has opened. 

With reference to the forces acting on the parachutist, explain why the graph has this shape in the region marked. 

The quality of written communication will be assessed in your answer.

Calculate the maximum deceleration of the parachutist in the region of the graph marked D, which shows how the speed changes just after the parachute has opened. Show your method clearly.

Use the graph to find the total vertical distance fallen by the parachutist in the first 10 s of the jump. Show your method clearly.

During his descent, the parachutist drifts sideways in the wind and hits the ground with a vertical speed of 5.0 m s^–1 and a horizontal speed of 2.0 m s^–1. 

Calculate the resultant speed with which he hits the ground and the angle his resultant velocity makes with the vertical.