3.02d Constant acceleration: SUVAT formulae

6 In this question you should take \(\boldsymbol { g \) to be \(\mathbf { 1 0 } \mathrm { ms } ^ { \boldsymbol { - } \mathbf { 2 } }\).} Piran finds a disused mineshaft on his land and wants to know its depth, \(d\) metres.
Local records state that the mineshaft is between 150 and 200 metres deep.
He drops a small stone down the mineshaft and records the time, \(T\) seconds, until he hears it hit the bottom. It takes 8.0 seconds. Piran tries three models, \(\mathrm { A } , \mathrm { B }\) and C .
In model A, Piran uses the formula \(d = 5 T ^ { 2 }\) to estimate the depth.

1. Find the depth that model A gives and comment on whether it is consistent with the local records. Explain how the formula in model A is obtained. In model B, Piran uses the speed-time graph in Fig. 6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-5_762_1176_1087_424} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure}
2. Calculate the depth of the mineshaft according to model B. Comment on whether this depth is consistent with the local records.
3. Describe briefly one respect in which model B is the same as model A and one respect in which it is different. Piran then tries model C in which the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by $$\begin{aligned} & v = 10 t - t ^ { 2 } \text { for } 0 \leqslant t \leqslant 5 \\ & v = 25 \text { for } 5 < t \leqslant 8 \end{aligned}$$
4. Calculate the depth of the mineshaft according to model C. Comment on whether this depth is consistent with the local records.
5. Describe briefly one respect in which model C is similar to model B and one respect in which it is different.

2 In this question the value of \(g\) should be taken as \(10 \mathrm {~m \mathrm {~s} ^ { 2 }\).} As shown in Fig. 8, particles A and B are projected towards one another. Each particle has an initial speed of \(10 \mathrm {~m} \mathrm {~s} ^ { 1 }\) vertically and \(20 \mathrm {~m} \mathrm {~s} { } ^ { 1 }\) horizontally. Initially A and B are 70 m apart horizontally and B is 15 m higher than A . Both particles are projected over horizontal ground. \begin{figure}[h] \end{figure}

1. Show that, \(t\) seconds after projection, the height in metres of each particle above its point of projection is \(10 t - 5 t ^ { 2 }\).
2. Calculate the horizontal range of A . Deduce that A hits the horizontal ground between the initial positions of A and B .
3. Calculate the horizontal distance travelled by B before reaching the ground.
4. Show that the paths of the particles cross but that the particles do not collide if they are projected at the same time. In fact, particle A is projected 2 seconds after particle B .
5. Verify that the particles collide 0.75 seconds after A is projected.

7 \includegraphics[max width=\textwidth, alt={}, center]{28beb431-45d5-4300-88fe-00d05d78790b-07_512_1072_484_502} The diagram shows the velocity-time graph for a train travelling on a straight level track between stations \(A\) and \(B\) that are 2 km apart. The train leaves \(A\), accelerating uniformly from rest for 400 m until reaching a speed of \(32 \mathrm {~ms} ^ { - 1 }\). The train then travels at this steady speed for \(T\) seconds before decelerating uniformly at \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\). Find the total time for the journey.

3. \begin{figure}[h] \end{figure} A sprinter runs a race of 200 m . Her total time for running the race is 25 s . Figure 2 is a sketch of the speed-time graph for the motion of the sprinter. She starts from rest and accelerates uniformly to a speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 s . The speed of \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is maintained for 16 s and she then decelerates uniformly to a speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of the race. Calculate

1. the distance covered by the sprinter in the first 20 s of the race,
2. the value of \(u\),
3. the deceleration of the sprinter in the last 5 s of the race.

5. Figure 4 \includegraphics[max width=\textwidth, alt={}, center]{94d9432d-1723-4549-ad5e-d4be0f5fd083-009_609_1026_301_516} A block of wood \(A\) of mass 0.5 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a small smooth pulley \(P\) fixed at the edge of the table. The other end of the string is attached to a ball \(B\) of mass 0.8 kg which hangs freely below the pulley, as shown in Figure 4. The coefficient of friction between \(A\) and the table is \(\mu\). The system is released from rest with the string taut. After release, \(B\) descends a distance of 0.4 m in 0.5 s . Modelling \(A\) and \(B\) as particles, calculate

1. the acceleration of \(B\),
2. the tension in the string,
3. the value of \(\mu\).
4. State how in your calculations you have used the information that the string is inextensible.

5 A golf ball is projected from a point \(O\) with initial velocity \(V\) at an angle \(\alpha\) to the horizontal. The ball first hits the ground at a point \(A\) which is at the same horizontal level as \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-005_227_602_484_735} It is given that \(V \cos \alpha = 6 u\) and \(V \sin \alpha = 2.5 u\).

1. Show that the time taken for the ball to travel from \(O\) to \(A\) is \(\frac { 5 u } { g }\).
2. Find, in terms of \(g\) and \(u\), the distance \(O A\).
3. Find \(V\), in terms of \(u\).
4. State, in terms of \(u\), the least speed of the ball during its flight from \(O\) to \(A\).

6 A van moves from rest on a straight horizontal road.

1. In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   During the second stage, the van accelerates from \(4 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
   During the third stage, the van accelerates from \(12 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\).
   1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
   2. Find the total distance that the van travels during the 30 seconds.
   3. Find the average speed of the van during the 30 seconds.
   4. Find the greatest acceleration of the van during the 30 seconds.
2. In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below. \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-006_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
   1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
   2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).

3

1. A small stone is dropped from a height of 25 metres above the ground.
   1. Find the time taken for the stone to reach the ground.
   2. Find the speed of the stone as it reaches the ground.
2. A large package is dropped from the same height as the stone. Explain briefly why the time taken for the package to reach the ground is likely to be different from that for the stone.
   (2 marks)

5 A golf ball is projected from a point \(O\) with initial velocity \(V\) at an angle \(\alpha\) to the horizontal. The ball first hits the ground at a point \(A\) which is at the same horizontal level as \(O\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-4_227_602_484_735} It is given that \(V \cos \alpha = 6 u\) and \(V \sin \alpha = 2.5 u\).

1. Show that the time taken for the ball to travel from \(O\) to \(A\) is \(\frac { 5 u } { g }\).
2. Find, in terms of \(g\) and \(u\), the distance \(O A\).
3. Find \(V\), in terms of \(u\).
4. State, in terms of \(u\), the least speed of the ball during its flight from \(O\) to \(A\).

6 A van moves from rest on a straight horizontal road.

1. In a simple model, the first 30 seconds of the motion are represented by three separate stages, each lasting 10 seconds and each with a constant acceleration. During the first stage, the van accelerates from rest to a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   During the second stage, the van accelerates from \(4 \mathrm {~ms} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\).
   During the third stage, the van accelerates from \(12 \mathrm {~ms} ^ { - 1 }\) to \(16 \mathrm {~ms} ^ { - 1 }\).
   1. Sketch a velocity-time graph to represent the motion of the van during the first 30 seconds of its motion.
   2. Find the total distance that the van travels during the 30 seconds.
   3. Find the average speed of the van during the 30 seconds.
   4. Find the greatest acceleration of the van during the 30 seconds.
2. In another model of the 30 seconds of the motion, the acceleration of the van is assumed to vary during the first and third stages of the motion, but to be constant during the second stage, as shown in the velocity-time graph below. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-5_554_1138_1432_539} The velocity of the van takes the same values at the beginning and the end of each stage of the motion as in part (a).
   1. State, with a reason, whether the distance travelled by the van during the first 10 seconds of the motion in this model is greater or less than the distance travelled during the same time interval in the model in part (a).
   2. Give one reason why this model represents the motion of the van more realistically than the model in part (a).

2 A sprinter accelerates from rest at a constant rate for the first 10 metres of a 100 -metre race. He takes 2.5 seconds to run the first 10 metres.

1. Find the acceleration of the sprinter during the first 2.5 seconds of the race.
2. Show that the speed of the sprinter at the end of the first 2.5 seconds of the race is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. The sprinter completes the 100 -metre race, travelling the remaining 90 metres at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the total time taken for the sprinter to travel the 100 metres.
4. Calculate the average speed of the sprinter during the 100 -metre race.

4 A ball is released from rest at a height of 15 metres above ground level.

1. Find the speed of the ball when it hits the ground, assuming that no air resistance acts on the ball.
2. In fact, air resistance does act on the ball. Assume that the air resistance force has a constant magnitude of 0.9 newtons. The ball has a mass of 0.5 kg .
   1. Draw a diagram to show the forces acting on the ball, including the magnitudes of the forces acting.
   2. Show that the acceleration of the ball is \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   3. Find the speed at which the ball hits the ground.
   4. Explain why the assumption that the air resistance force is constant may not be valid.

1 A ball is released from rest at a height \(h\) metres above ground level. The ball hits the ground 1.5 seconds after it is released. Assume that the ball is a particle that does not experience any air resistance.

1. Show that the speed of the ball is \(14.7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it hits the ground.
2. Find \(h\).
3. Find the distance that the ball has fallen when its speed is \(5 \mathrm {~ms} ^ { - 1 }\).

6 A box, of mass 3 kg , is placed on a slope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The box slides down the slope. Assume that air resistance can be ignored.

1. A simple model assumes that the slope is smooth.
   1. Draw a diagram to show the forces acting on the box.
   2. Show that the acceleration of the box is \(4.9 \mathrm {~ms} ^ { - 2 }\).
2. A revised model assumes that the slope is rough. The box slides down the slope from rest, travelling 5 metres in 2 seconds.
   1. Show that the acceleration of the box is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the magnitude of the friction force acting on the box.
   3. Find the coefficient of friction between the box and the slope.
   4. In reality, air resistance affects the motion of the box. Explain how its acceleration would change if you took this into account.

17 A ball is released from a great height so that it falls vertically downwards towards the surface of the Earth. 17

1. Using a simple model, Andy predicts that the velocity of the ball, exactly 2 seconds after being released from rest, is \(2 g \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Show how Andy has obtained his prediction.
   17
2. Using a refined model, Amy predicts that the ball's acceleration, \(a \mathrm {~ms} ^ { - 2 }\), at time \(t\) seconds after being released from rest is $$a = g - 0.1 v$$ where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the ball at time \(t\) seconds. Find an expression for \(v\) in terms of \(t\).
   17
3. Comment on the value of \(v\) for the two models as \(t\) becomes large.

12 A car is travelling along a straight horizontal road with initial velocity \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The car begins to accelerate at a constant rate \(a \mathrm {~ms} ^ { - 2 }\) for 5 seconds, to reach a final velocity of \(4 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Express \(a\) in terms of \(u\).
Circle your answer.
[0pt] [1 mark] \(a = 0.2 u\) \(a = 0.4 u\) \(a = 0.6 u\) \(a = 0.8 u\)

19

1. The tension in the rope is 230 N
   The crate accelerates up the ramp at \(1.2 \mathrm {~ms} ^ { - 2 }\) Find the coefficient of friction between the crate and the ramp.
   19
2. (i) The crate takes 3.8 seconds to reach the top of the ramp.
   Find the distance \(O A\).
   [0pt] [3 marks]
   19 (b) (ii) Other than air resistance, state one assumption you have made about the crate in answering part (b)(i). \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-32_2492_1721_217_150}

13 A ball falls freely towards the Earth.
The ball passes through two different fixed points \(M\) and \(N\) before reaching the Earth's surface. At \(M\) the ball has velocity \(u \mathrm {~ms} ^ { - 1 }\) At \(N\) the ball has velocity \(3 u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) It can be assumed that:

• the motion is due to gravitational force only
• the acceleration due to gravity remains constant throughout.

13

1. Show that the time taken for the ball to travel from \(M\) to \(N\) is \(\frac { 2 u } { g }\) seconds.
   [0pt] [2 marks] 13
2. Point \(M\) is \(h\) metres above the Earth. Show that \(h > \frac { 4 u ^ { 2 } } { g }\) Fully justify your answer.
   The car is moving in a straight line.
   The acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the car at time \(t\) seconds is given by $$a = 3 k t ^ { 2 } - 2 k t + 1$$ where \(k\) is a constant.
   When \(t = 3\) the car has a velocity of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Show that \(k = \frac { 1 } { 3 }\)

19 A wooden toy comprises a train engine and a trailer connected to each other by a light, inextensible rod. The train engine has a mass of 1.5 kilograms.
The trailer has a mass 0.7 kilograms.
A string inclined at an angle of \(40 ^ { \circ }\) above the horizontal is attached to the front of the train engine. The tension in the string is 2 newtons.
As a result the toy moves forward, from rest, in a straight line along a horizontal surface with acceleration \(0.06 \mathrm {~ms} ^ { - 2 }\) as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-30_373_789_904_756} As it moves the train engine experiences a total resistance force of 0.8 N
19

1. Show that the total resistance force experienced by the trailer is approximately 0.6 N
   19
2. At the instant that the toy reaches a speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks. As a result of this the train engine and trailer decelerate at a constant rate until they come to rest, having travelled a distance of \(h\) metres. It can be assumed that the resistance forces remain unchanged.
   19 (b) (i) Find the tension in the rod after the string has broken.
   

   19 (b) (ii) Find \(h\)

   Do not write outside the box

   \includegraphics[max width=\textwidth, alt={}]{de8a7d38-a665-4feb-854e-ac83f413d133-33_2488_1716_219_153}
   Nell and her pet dog Maia are visiting the beach.
   The beach surface can be assumed to be level and horizontal. Nell and Maia are initially standing next to each other.
   Nell throws a ball forward, from a height of 1.8 metres above the surface of the beach, at an angle of \(60 ^ { \circ }\) above the horizontal with a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Exactly 0.2 seconds after the ball is thrown, Maia sets off from Nell and runs across the surface of the beach, in a straight line with a constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Maia catches the ball when it is 0.3 metres above ground level as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-34_778_1287_1027_463}

1. A man throws a tennis ball into the air so that, at the instant when the ball leaves his hand, the ball is 2 m above the ground and is moving vertically upwards with speed \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

The motion of the ball is modelled as that of a particle moving freely under gravity and the acceleration due to gravity is modelled as being of constant magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The ball hits the ground \(T\) seconds after leaving the man's hand.
Using the model, find the value of \(T\).

1. A train travels along a straight horizontal track between two stations, \(A\) and \(B\).

In a model of the motion, the train starts from rest at \(A\) and moves with constant acceleration \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 80 s .
The train then moves at constant velocity before it moves with a constant deceleration of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).

1. For this model of the motion of the train between \(A\) and \(B\),
   1. state the value of the constant velocity of the train,
   2. state the time for which the train is decelerating,
   3. sketch a velocity-time graph. The total distance between the two stations is 4800 m .
2. Using the model, find the total time taken by the train to travel from \(A\) to \(B\).
3. Suggest one improvement that could be made to the model of the motion of the train from \(A\) to \(B\) in order to make the model more realistic.

1. A car is moving along a straight horizontal road with constant acceleration. There are three points \(A , B\) and \(C\), in that order, on the road, where \(A B = 22 \mathrm {~m}\) and \(B C = 104 \mathrm {~m}\). The car takes 2 s to travel from \(A\) to \(B\) and 4 s to travel from \(B\) to \(C\).

Find

1. the acceleration of the car,
2. the speed of the car at the instant it passes \(A\).

1. The points \(A\) and \(B\) lie on the same straight horizontal road.

Figure 2, on page 11, shows the speed-time graph of a cyclist \(P\), for his journey from \(A\) to \(B\).
At time \(t = 0 , P\) starts from rest at \(A\) and accelerates uniformly for 9 seconds until his speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) He then travels at constant speed \(V \mathrm {~ms} ^ { - 1 }\) When \(t = 42\), cyclist \(P\) passes \(B\).
Given that the distance \(A B\) is 120 m ,

1. show that \(V = 3.2\)
2. Find the acceleration of cyclist \(P\) between \(t = 0\) and \(t = 9\) Cyclist \(P\) continues to cycle along the road in the same direction at the same constant speed, \(V \mathrm {~ms} ^ { - 1 }\) When \(t = 6\), a second cyclist \(Q\) sets off from \(A\) and travels in the same direction as \(P\) along the same road. She accelerates for \(T\) seconds until her speed is \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) She then travels at constant speed \(3.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Cyclist \(Q\) catches up with \(P\) when \(t = 54\)
3. On Figure 2, on page 11, sketch a speed-time graph showing the journeys of both cyclists, for the interval \(0 \leqslant t \leqslant 54\)
4. Find the value of \(T\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2f2f89a6-cec4-444d-95d9-0112887d87eb-11_661_1509_292_278} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure} A copy of Figure 2 is on page 13 if you need to redraw your answer to part (c). Only use this copy of Figure 2 if you need to redraw your answer to part (c). \includegraphics[max width=\textwidth, alt={}, center]{2f2f89a6-cec4-444d-95d9-0112887d87eb-13_666_1509_374_278} \section*{Copy of Figure 2}

5. \begin{figure}[h] \end{figure} Two particles, \(P\) and \(Q\), have masses 3 kg and 5 kg respectively. The particles are connected by a light inextensible string which passes over a small smooth fixed pulley. The particles are released from rest with the string taut and the hanging parts of the string vertical, as shown in Figure 3. Immediately after the particles are released from rest, \(P\) moves upwards with acceleration \(a \mathrm {~ms} ^ { - 2 }\) and the tension in the string is \(T\) newtons.

1. Write down an equation of motion for \(P\).
2. Find the value of \(T\). The total force acting on the pulley due to the string has magnitude \(F\) newtons.
3. Find the value of \(F\). Initially, \(Q\) is 10 m above horizontal ground and \(P\) is more than 2 m below the pulley.
   At the instant when \(Q\) has descended a distance of 2 m , the string breaks and \(Q\) falls to the ground.
4. Find the speed of \(Q\) at the instant it hits the ground.