Motion graphs

Figure 2 shows a speed‑time graph for a model of the motion of an athlete running a 200 m race in 24 s. The athlete

• starts from rest at time t = 0 and accelerates at a constant rate, reaching a speed of

• then moves at a constant speed of 10 m s^–1 fromt = 4 tot = 18

• then decelerates at a constant rate from t = 18 to t = 24, crossing the finishing line

Using the model,

find the acceleration of the athlete during the first 4 s of the race, stating the units of your answer,

find the distance covered by the athlete during the first 18 s of the race,

find the value of U.

A small stone A of mass 3m is attached to one end of a string. A small stone B of mass m is attached to the other end of the string. Initially A is held at rest on a fixed rough plane. The plane is inclined to the horizontal at an angle α, where tanα = The string passes over a pulley P that is fixed at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane. Stone B hangs freely below P, as shown in Figure 1. The coefficient of friction between A and the plane is Stone A is released from rest and begins to move down the plane. The stones are modelled as particles. The pulley is modelled as being small and smooth. The string is modelled as being light and inextensible.

Using the model for the motion of the system before B reaches the pulley,

write down an equation of motion for A

show that the acceleration of A is g

sketch a velocity-time graph for the motion of B, from the instant when A is released from rest to the instant just before B reaches the pulley, explaining your answer. In reality, the string is not light.

State how this would affect the working in part (b).

A small stone A of mass 3m is attached to one end of a string.

A small stone B of mass m is attached to the other end of the string.

Initially A is held at rest on a fixed rough plane. The plane is inclined to the horizontal at an angle α, where tanα =

The string passes over a pulley P that is fixed at the top of the plane.

The part of the string from A to P is parallel to a line of greatest slope of the plane.

Stone B hangs freely below P, as shown in Figure 1. The coefficient of friction between A and the plane is

Stone A is released from rest and begins to move down the plane.

The stones are modelled as particles.

The pulley is modelled as being small and smooth.

The string is modelled as being light and inextensible.

Using the model for the motion of the system before B reaches the pulley,

write down an equation of motion for A

show that the acceleration of A is g

sketch a velocity-time graph for the motion of B, from the instant when A is released from rest to the instant just before B reaches the pulley, explaining your answer.

In reality, the string is not light.

State how this would affect the working in part (b).

A ball of mass 0.2 kg is projected vertically downwards with speed U m s^–1 from a point A which is 2.5 m above horizontal ground. The ball hits the ground. Immediately after hitting the ground, the ball rebounds vertically with a speed of 10 m s^–1. The ball receives an impulse of magnitude 7 N s in its impact with the ground. By modelling the ball as a particle and ignoring air resistance, find

the value of U.

After hitting the ground, the ball moves vertically upwards and passes through a point B which is 1 m above the ground.

Find the time between the instant when the ball hits the ground and the instant when the ball first passes through B.

Sketch a velocity-time graph for the motion of the ball from when it was projected from A to when it first passes through B. (You need not make any further calculations to draw this sketch.)

Two trains M and N are moving in the same direction along parallel straight horizontal tracks. At time t = 0, M overtakes N whilst they are travelling with speeds 40 m s^–1 and 30 m s^–1 respectively. Train M overtakes train N as they pass a point X at the side of the tracks . After overtaking N, train M maintains its speed of 40 m s^–1 for T seconds and then decelerates uniformly, coming to rest next to a point Y at the side of the tracks. After being overtaken, train N maintains its speed of 30 m s^–1 for 25 s and then decelerates uniformly, also coming to rest next to the point Y. The times taken by the trains to travel between X and Y are the same.

Sketch, on the same diagram, the speed-time graphs for the motions of the two trains between X and Y.

Given that XY = 975 m,

find the value of T.

A car starts from rest and moves with constant acceleration along a straight horizontal road. The car reaches a speed of V m s^–1 in 20 seconds. It moves at constant speed V m s^–1 for the next 30 seconds, then moves with constant deceleration m s^–2 until it has speed 8 m s^–1. It moves at speed 8 m s^–1 for the next 15 seconds and then moves with constant deceleration m s^–2 until it comes to rest.

Sketch, in the space below, a speed-time graph for this journey.

In the first 20 seconds of this journey the car travels 140 m. Find

the value of V,

the total time for this journey,

the total distance travelled by the car.

A car is travelling along a straight horizontal road. The car takes 120 s to travel between two sets of traffic lights which are 2145 m apart. The car starts from rest at the first set of traffic lights and moves with constant acceleration for 30 s until its speed is 22 m s^–1. The car maintains this speed for T seconds. The car then moves with constant deceleration, coming to rest at the second set of traffic lights.

Sketch, in the space below, a speed-time graph for the motion of the car between the two sets of traffic lights.

Find the value of T.

A motorcycle leaves the first set of traffic lights 10 s after the car has left the first set of traffic lights. The motorcycle moves from rest with constant acceleration, a m s^–2, and passes the car at the point A which is 990 m from the first set of traffic lights. When the motorcycle passes the car, the car is moving with speed 22 m s^–1.

Find the time it takes for the motorcycle to move from the first set of traffic lights to the point A.

Find the value of a.

The velocity-time graph in Figure 4 represents the journey of a train P travelling along a straight horizontal track between two stations which are 1.5 km apart. The train P leaves the first station, accelerating uniformly from rest for 300 m until it reaches a speed of 30 m s^–1. The train then maintains this speed for T seconds before decelerating uniformly at 1.25 m s^–2, coming to rest at the next station.

Find the acceleration of P during the first 300 m of its journey.

Find the value of T.

A second train Q completes the same journey in the same total time. The train leaves the first station, accelerating uniformly from rest until it reaches a speed of V m s^–1 and then immediately decelerates uniformly until it comes to rest at the next station.

Sketch on the diagram above, a velocity-time graph which represents the journey of train Q.

Find the value of V.

A car is moving on a straight horizontal road. At time t = 0, the car is moving with speed 20 m s^–1 and is at the point A. The car maintains the speed of 20 m s^–1 for 25 s. The car

8 m s^–1. The car then moves with constant speed 8 m s^–1 for 60 s. The car then moves with constant acceleration until it is moving with speed 20 m s^–1 at the point B.

Sketch a speed-time graph to represent the motion of the car from A to B.

Find the time for which the car is decelerating.

Given that the distance from A to B is 1960 m,

find the time taken for the car to move from A to B.

A girl runs a 400 m race in a time of 84 s. In a model of this race, it is assumed that, starting from rest, she moves with constant acceleration for 4 s, reaching a speed of 5 m s^–1. She maintains this speed for 60 s and then moves with constant deceleration for 20 s, crossing the finishing line with a speed of V m s^–1.

Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race.

Find the distance run by the girl in the first 64 s of the race.

Find the value of V.

Find the deceleration of the girl in the final 20 s of her race.

A car accelerates uniformly from rest for 20 seconds. It moves at constant speed v m ss¹ for the next 40 seconds and then decelerates uniformly for 10 seconds until it comes to rest.

For the motion of the car, sketch

a speed-time graph,

an acceleration-time graph.

Given that the total distance moved by the car is 880 m,

find the value of v.

Two cars P and Q are moving in the same direction along the same straight horizontal road. Car P is moving with constant speed 25 m s¹. At time t 0, P overtakes Q which is moving with constant speed 20 m s¹. From t T seconds, P decelerates uniformly, coming to rest at a point X which is 800 m from the point where P overtook Q. From t 25 s, Q decelerates uniformly, coming to rest at the same point X at the same instant as P.

Sketch, on the same axes, the speed-time graphs of the two cars for the period from t 0 to the time when they both come to rest at the point X.

Find the value of T.

An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of 8 m s^–1. This speed is then maintained for T seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s.

In the space below, sketch a speed-time graph to illustrate the motion of the athlete.

Calculate the value of T.

A car is moving along a straight horizontal road. The speed of the car as it passes the point A is 25 m s^–1 and the car maintains this speed for 30 s. The car then decelerates uniformly to a speed of 10 m s^–1. The speed of 10 m s^–1 is then maintained until the car passes the point B. The time taken to travel from A to B is 90 s and AB = 1410 m.

Sketch, in the space below, a speed-time graph to show the motion of the car from A to B.

Calculate the deceleration of the car as it decelerates from 25 m s^–1 to 10 m s^–1.

A car moves along a horizontal straight road, passing two points A and B. At A the speed of the car is 15 m s^–1. When the driver passes A, he sees a warning sign W ahead of him, 120 m away. He immediately applies the brakes and the car decelerates with uniform deceleration, reaching W with speed 5 m s^–1. At W, the driver sees that the road is clear. He then immediately accelerates the car with uniform acceleration for 16 s to reach a speed of V m s^–1 (V > 15). He then maintains the car at a constant speed of V m s^–1. Moving at this constant speed, the car passes B after a further 22 s.

Sketch, in the space below, a speed-time graph to illustrate the motion of the car as it moves from A to B.

Find the time taken for the car to move from A to B.

The distance from A to B is 1 km.

Find the value of V.