3.02d Constant acceleration: SUVAT formulae

9 In an experiment, a small box is hit across a floor. After it has been hit, the box slides without rotation. The box passes a point A. The distance the box travels after passing A before coming to rest is \(S\) metres and the time this takes is \(T\) seconds. The only resistance to the box's motion is friction due to the floor. The mass of the box is \(m \mathrm {~kg}\) and the frictional force is a constant \(F\).

1. 1. Find the equation of motion for the box while it is sliding.
   2. Show that \(S = k T ^ { 2 }\) where \(k = \frac { F } { 2 m }\).
2. Given that \(k = 1.4\), find the value of the coefficient of friction between the box and the floor.

1 A train travels along a straight horizontal track. It is travelling at a speed of \(12 \mathrm {~ms} ^ { - 1 }\) when it begins to accelerate uniformly. It reaches a speed of \(40 \mathrm {~ms} ^ { - 1 }\) after accelerating for 100 seconds.

1. 1. Show that the acceleration of the train is \(0.28 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the distance that the train travelled in the 100 seconds.
2. The mass of the train is 200 tonnes and a resistance force of 40000 N acts on the train. Find the magnitude of the driving force produced by the engine that acts on the train as it accelerates.

8 A football is placed on a horizontal surface. It is then kicked, so that it has an initial velocity of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal.

1. State two modelling assumptions that it would be appropriate to make when considering the motion of the football.
2. 1. Find the time that it takes for the ball to reach its maximum height.
   2. Hence show that the maximum height of the ball is 3.04 metres, correct to three significant figures.
3. After the ball has reached its maximum height, it hits the bar of a goal at a height of 2.44 metres. Find the horizontal distance of the goal from the point where the ball was kicked.

2 A lift rises vertically from rest with a constant acceleration.
After 4 seconds, it is moving upwards with a velocity of \(2 \mathrm {~ms} ^ { - 1 }\).
It then moves with a constant velocity for 5 seconds.
The lift then slows down uniformly, coming to rest after it has been moving for a total of 12 seconds.

1. Sketch a velocity-time graph for the motion of the lift.
2. Calculate the total distance travelled by the lift.
3. The lift is raised by a single vertical cable. The mass of the lift is 300 kg . Find the maximum tension in the cable during this motion.

4 The diagram shows a block, of mass 13 kg , on a rough horizontal surface. It is attached by a string that passes over a smooth peg to a sphere of mass 7 kg , as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-3_323_974_1256_575} The system is released from rest, and after 4 seconds the block and the sphere both have speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and the block has not reached the peg.

1. State two assumptions that you should make about the string in order to model the motion of the sphere and the block.
2. Show that the acceleration of the sphere is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the tension in the string.
4. Find the coefficient of friction between the block and the surface.

5 A girl in a boat is rowing across a river, in which the water is flowing at \(0.1 \mathrm {~ms} ^ { - 1 }\). The velocity of the boat relative to the water is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is perpendicular to the bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-4_314_1152_468_450}

1. Find the magnitude of the resultant velocity of the boat.
2. Find the acute angle between the resultant velocity and the bank.
3. The width of the river is 15 metres.
   1. Find the time that it takes the boat to cross the river.
   2. Find the total distance travelled by the boat as it crosses the river.

7 A golf ball is struck from a point on horizontal ground so that it has an initial velocity of \(50 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal. Assume that the golf ball is a particle and its weight is the only force that acts on it once it is moving.

1. Find the maximum height of the golf ball.
2. After it has reached its maximum height, the golf ball descends but hits a tree at a point which is at a height of 6 metres above ground level. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-5_289_1358_813_335} \begin{displayquote} Find the time that it takes for the ball to travel from the point where it was struck to the tree. \end{displayquote}

1 A crane is used to lift a crate, of mass 70 kg , vertically upwards. As the crate is lifted, it accelerates uniformly from rest, rising 8 metres in 5 seconds.

1. Show that the acceleration of the crate is \(0.64 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. The crate is attached to the crane by a single cable. Assume that there is no resistance to the motion of the crate. Find the tension in the cable.
3. Calculate the average speed of the crate during these 5 seconds.

5 A puck, of mass 0.2 kg , is placed on a slope inclined at \(20 ^ { \circ }\) above the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-3_280_773_1249_623} The puck is hit so that initially it moves with a velocity of \(4 \mathrm {~ms} ^ { - 1 }\) directly up the slope.

1. A simple model assumes that the surface of the slope is smooth.
   1. Show that the acceleration of the puck up the slope is \(- 3.35 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to three significant figures.
   2. Find the distance that the puck will travel before it comes to rest.
   3. What will happen to the puck after it comes to rest? Explain why.
2. A revised model assumes that the surface is rough and that the coefficient of friction between the puck and the surface is 0.5 .
   1. Show that the magnitude of the friction force acting on the puck during this motion is 0.921 N , correct to three significant figures.
   2. Find the acceleration of the puck up the slope.
   3. What will happen to the puck after it comes to rest in this case? Explain why.

7 A golfer hits a ball which is on horizontal ground. The ball initially moves with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) above the horizontal. There is a pond further along the horizontal ground. The diagram below shows the initial position of the ball and the position of the pond. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-5_387_1230_502_395}

1. State two assumptions that you should make in order to model the motion of the ball.
   (2 marks)
2. Show that the horizontal distance, in metres, travelled by the ball when it returns to ground level is $$\frac { V ^ { 2 } \sin 40 ^ { \circ } \cos 40 ^ { \circ } } { 4.9 }$$
3. Find the range of values of \(V\) for which the ball lands in the pond.

3 A box of mass 4 kg is held at rest on a plane inclined at an angle of \(40 ^ { \circ }\) to the horizontal. The box is then released and slides down the plane.

1. A simple model assumes that the only forces acting on the box are its weight and the normal reaction from the plane. Show that, according to this simple model, the acceleration of the box would be \(6.30 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to three significant figures.
2. In fact, the box moves down the plane with constant acceleration and travels 0.9 metres in 0.6 seconds. By using this information, find the acceleration of the box.
3. Explain why the answer to part (b) is less than the answer to part (a).

4 Two particles, \(A\) and \(B\), are connected by a string that passes over a fixed peg, as shown in the diagram. The mass of \(A\) is 9 kg and the mass of \(B\) is 11 kg .
The particles are released from rest in the position shown, where \(B\) is \(d\) metres higher than \(A\). The motion of the particles is to be modelled using simple assumptions.

1. State one assumption that should be made about the peg.
2. State two assumptions that should be made about the string.
3. By forming an equation of motion for each of the particles \(A\) and \(B\), show that the acceleration of each particle has magnitude \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
4. When the particles have been moving for 0.5 seconds, they are at the same level.
   1. Find the speed of the particles at this time.
   2. Find \(d\).

2 The graph shows how the velocity of a train varies as it moves along a straight railway line. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-04_574_1595_402_203}

1. Find the total distance travelled by the train.
2. Find the average speed of the train.
3. Find the acceleration of the train during the first 10 seconds of its motion.
4. The mass of the train is 200 tonnes. Find the magnitude of the resultant force acting on the train during the first 10 seconds of its motion.

3 A car, of mass 1200 kg , tows a caravan, of mass 1000 kg , along a straight horizontal road. The caravan is attached to the car by a horizontal tow bar, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-06_277_901_484_584} Assume that a constant resistance force of magnitude 200 newtons acts on the car and a constant resistance force of magnitude 300 newtons acts on the caravan. A constant driving force of magnitude \(P\) newtons acts on the car in the direction of motion. The car and caravan accelerate at \(0.8 \mathrm {~ms} ^ { - 2 }\).

1. 1. Find \(P\).
   2. Find the magnitude of the force in the tow bar that connects the car to the caravan.
2. 1. Find the time that it takes for the speed of the car and caravan to increase from \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   2. Find the distance that they travel in this time.
3. Explain why the assumption that the resistance forces are constant is unrealistic.
   (1 mark)

3 The diagram shows a velocity-time graph for a train as it moves on a straight horizontal track for 50 seconds. \includegraphics[max width=\textwidth, alt={}, center]{d42b2e88-74ea-486b-bb47-f512eb0c185d-3_620_1221_408_358}

1. Find the distance that the train moves in the first 28 seconds.
2. Calculate the total distance moved by the train during the 50 seconds.
3. Hence calculate the average speed of the train.
4. Find the displacement of the train from its initial position when it has been moving for 50 seconds.
5. Hence calculate the average velocity of the train.
6. Find the acceleration of the train in the first 18 seconds of its motion.

6 A cyclist freewheels, with a constant acceleration, in a straight line down a slope. As the cyclist moves 50 metres, his speed increases from \(4 \mathrm {~ms} ^ { - 1 }\) to \(10 \mathrm {~ms} ^ { - 1 }\).

1. 1. Find the acceleration of the cyclist.
   2. Find the time that it takes the cyclist to travel this distance.
2. The cyclist has a mass of 70 kg . Calculate the magnitude of the resultant force acting on the cyclist.
3. The slope is inclined at an angle \(\alpha\) to the horizontal.
   1. Find \(\alpha\) if it is assumed that there is no resistance force acting on the cyclist.
   2. Find \(\alpha\) if it is assumed that there is a constant resistance force of magnitude 30 newtons acting on the cyclist.
4. Make a criticism of the assumption described in part (c)(ii).

1 A car travels on a straight horizontal race track. The car decelerates uniformly from a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it travels a distance of 640 metres. The car then accelerates uniformly, travelling a further 1820 metres in 70 seconds.

1. 1. Find the time that it takes the car to travel the first 640 metres.
   2. Find the deceleration of the car during the first 640 metres.
2. 1. Find the acceleration of the car as it travels the further 1820 metres.
   2. Find the speed of the car when it has completed the further 1820 metres.
3. Find the average speed of the car as it travels the 2460 metres.

8 A golf ball is hit from a point on a horizontal surface, so that it has an initial velocity \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal. The ball travels through the air and after 2.4 seconds hits a vertical wall at a height of 3 metres. The wall is at a horizontal distance of 38.4 metres from the point where the ball was hit. The path of the ball is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ccc1db66-9700-4f22-905e-cc0bdf1fd3c1-18_300_1000_566_520} Assume that the weight of the ball is the only force that acts on it as it travels through the air.

1. Find the horizontal component of the velocity of the ball.
2. \(\quad\) Find \(V\).
3. \(\quad\) Find \(\alpha\).

4 Two particles, \(A\) of mass 5 kg and \(B\) of mass 9 kg , are attached to the ends of a light inextensible string. The string passes over a light smooth pulley as shown in the diagram. The particles are released from rest at the same height. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-3_378_287_1580_872}

1. By forming an equation of motion for each particle, show that the magnitude of the acceleration of each particle is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the tension in the string.
3. When \(B\) has been moving for 0.5 seconds it hits the floor. Find the height of \(A\), above the floor, at this time. Assume that \(A\) is still below the pulley when \(B\) hits the floor.
   (4 marks)

6 A ball is hit from horizontal ground with velocity \(( 10 \mathbf { i } + 24.5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively.

1. State two assumptions that you should make about the ball in order to make predictions about its motion.
2. The path of the ball is shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_771_705_625}
   1. Show that the time of flight of the ball is 5 seconds.
   2. Find the range of the ball.
3. In fact the ball hits a vertical wall that is 20 metres from the initial position of the ball. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-5_351_403_1466_769} Find the height of the ball when it hits the wall.
4. If a heavier ball were projected in the same way, would your answers to part (b) of this question change? Explain why.

1 A stone is dropped from a high bridge and falls vertically.

1. Find the distance that the stone falls during the first 4 seconds of its motion.
2. Find the average speed of the stone during the first 4 seconds of its motion.
3. State one modelling assumption that you have made about the forces acting on the stone during the motion.

3 A car travels along a straight horizontal road. The motion of the car can be modelled as three separate stages. During the first stage, the car accelerates uniformly from rest to a velocity of \(10 \mathrm {~ms} ^ { - 1 }\) in 6 seconds. During the second stage, the car travels with a constant velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 4 seconds. During the third stage of the motion, the car travels with a uniform retardation of magnitude \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest.

1. Show that the time taken for the third stage of the motion is 12.5 seconds.
2. Sketch a velocity-time graph for the car during the three stages of the motion.
3. Find the total distance travelled by the car during the motion.
4. State one criticism of the model of the motion.

5 A small block \(P\) is attached to another small block \(Q\) by a light inextensible string. The block \(P\) rests on a rough horizontal surface and the string hangs over a smooth peg so that \(Q\) hangs freely, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-4_222_426_507_810} The block \(P\) has mass 0.4 kg and the coefficient of friction between \(P\) and the surface is 0.5 . The block \(Q\) has mass 0.3 kg . The system is released from rest and \(Q\) moves vertically downwards.

1. 1. Draw a diagram to show the forces acting on \(P\).
   2. Show that the frictional force between \(P\) and the surface has magnitude 1.96 newtons.
2. By forming an equation of motion for each block, show that the magnitude of the acceleration of each block is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the speed of the blocks after 3 seconds of motion.
4. After 3 seconds of motion, the string breaks. The blocks continue to move. Comment on how the speed of each block will change in the subsequent motion. For each block, give a reason for your answer.

7 A golf ball is struck from a point \(O\) with velocity \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(40 ^ { \circ }\) to the horizontal. The ball first hits the ground at a point \(P\), which is at a height \(h\) metres above the level of \(O\). \includegraphics[max width=\textwidth, alt={}, center]{cfe0bdbc-35e3-485f-a922-b652a72f4c95-5_318_990_484_543} The horizontal distance between \(O\) and \(P\) is 57 metres.

1. Show that the time that the ball takes to travel from \(O\) to \(P\) is 3.10 seconds, correct to three significant figures.
2. Find the value of \(h\).
3. 1. Find the speed with which the ball hits the ground at \(P\).
   2. Find the angle between the direction of motion and the horizontal as the ball hits the ground at \(P\).

7 A ball is hit by a bat so that, when it leaves the bat, its velocity is \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(35 ^ { \circ }\) above the horizontal. Assume that the ball is a particle and that its weight is the only force that acts on the ball after it has left the bat.

1. A simple model assumes that the ball is hit from the point \(A\) and lands for the first time at the point \(B\), which is at the same level as \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-4_321_1063_1370_484}
   1. Show that the time that it takes for the ball to travel from \(A\) to \(B\) is 4.68 seconds, correct to three significant figures.
   2. Find the horizontal distance from \(A\) to \(B\).
2. A revised model assumes that the ball is hit from the point \(C\), which is 1 metre above \(A\). The ball lands at the point \(D\), which is at the same level as \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{a381686b-0b1e-41ba-b88f-be1601e42098-4_431_1177_2181_420} Find the time that it takes for the ball to travel from \(C\) to \(D\).