3.02d Constant acceleration: SUVAT formulae

A particle \(P\) of mass \(0.5\) kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40°\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is \(0.6\).

1. Show that the magnitude of the frictional force acting on \(P\) is \(2.25\) N, correct to 3 significant figures. [3]
2. Find the acceleration of \(P\) when it is moving
   1. up the plane,
   2. down the plane.
   [4]
3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4\) m s\(^{-1}\).
   1. Find the length of time before \(P\) reaches its highest point.
   2. Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).
   [8]

A particle starts from rest at a point \(A\) at time \(t = 0\), where \(t\) is in seconds. The particle moves in a straight line. For \(0 \leq t \leq 4\) the acceleration is \(1.8t \text{ m s}^{-2}\), and for \(4 \leq t \leq 7\) the particle has constant acceleration \(7.2 \text{ m s}^{-2}\).

1. Find an expression for the velocity of the particle in terms of \(t\), valid for \(0 \leq t \leq 4\). [3]
2. Show that the displacement of the particle from \(A\) is 19.2 m when \(t = 4\). [4]
3. Find the displacement of the particle from \(A\) when \(t = 7\). [5]

\includegraphics{figure_6} The diagram shows the \((t, v)\) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for 25 s, coming to rest 8 m above ground level.

1. Find the greatest speed reached by the hoist during this stage. [2]

The second stage consists of a 40 s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest 40 m above ground level, arriving 135 s after leaving ground level. The hoist accelerates at \(0.02 \text{ m s}^{-2}\) for the first 40 s of the third stage, reaching a speed of \(V \text{ m s}^{-1}\). Find

1. the value of \(V\), [3]
2. the length of time during the third stage for which the hoist is moving at constant speed, [4]
3. the deceleration of the hoist in the final part of the third stage. [3]

A particle \(P\) of mass 0.5 kg moves upwards along a line of greatest slope of a rough plane inclined at an angle of \(40°\) to the horizontal. \(P\) reaches its highest point and then moves back down the plane. The coefficient of friction between \(P\) and the plane is 0.6.

1. Show that the magnitude of the frictional force acting on \(P\) is 2.25 N, correct to 3 significant figures. [3]
2. Find the acceleration of \(P\) when it is moving
   1. up the plane,
   2. down the plane.
   [4]
3. When \(P\) is moving up the plane, it passes through a point \(A\) with speed \(4 \text{ m s}^{-1}\).
   1. Find the length of time before \(P\) reaches its highest point.
   2. Find the total length of time for \(P\) to travel from the point \(A\) to its highest point and back to \(A\).
   [8]

The driver of a car accelerating uniformly from rest sees an obstruction. She brakes immediately bringing the car to rest with constant deceleration at a distance of \(6\) m from its starting point. The car travels in a straight line and is in motion for \(3\) seconds.

1. Sketch the \((t, v)\) graph for the car's motion. [2]
2. Calculate the maximum speed of the car during its motion. [3]
3. Hence, given that the acceleration of the car is \(2.4\) m s\(^{-2}\), calculate its deceleration. [4]

An object is projected vertically upwards with speed \(7\) m s\(^{-1}\). Calculate

1. the speed of the object when it is \(2.1\) m above the point of projection, [3]
2. the greatest height above the point of projection reached by the object, [3]
3. the time after projection when the object is travelling downwards with speed \(5.7\) m s\(^{-1}\). [3]

A stone is released from rest on a bridge and falls vertically into a lake. The stone has velocity \(14\text{ m s}^{-1}\) when it enters the lake.

1. Calculate the distance the stone falls before it enters the lake, and the time after its release when it enters the lake. [4]

The lake is \(15\text{ m}\) deep and the stone has velocity \(20\text{ m s}^{-1}\) immediately before it reaches the bed of the lake.

1. Given that there is no sudden change in the velocity of the stone when it enters the lake, find the acceleration of the stone while it is falling through the lake. [3]

A particle \(P\) is projected down a line of greatest slope on a smooth inclined plane. \(P\) has velocity \(5\text{ m s}^{-1}\) at the instant when it has been in motion for \(1.6\text{ s}\) and travelled a distance of \(6.4\text{ m}\). Calculate

1. the initial speed and the acceleration of \(P\), [5]
2. the inclination of the plane to the vertical. [3]

A particle is projected vertically upwards, from the ground, with a speed of \(28 \text{ m s}^{-1}\). Ignoring air resistance, find

1. the maximum height reached by the particle, [2]
2. the speed of the particle when it is 30 m above the ground, [3]
3. the time taken for the particle to fall from its highest point to a height of 30 m, [3]
4. the length of time for which the particle is more than 30 m above the ground. [2]

A helicopter rescue activity at sea is modelled as follows. The helicopter is stationary and a man is suspended from it by means of a vertical, light, inextensible wire that may be raised or lowered, as shown in Fig. 6.1. \includegraphics{figure_6_1}

1. When the man is descending with an acceleration 1.5 m s\(^{-2}\) downwards, how much time does it take for his speed to increase from 0.5 m s\(^{-1}\) downwards to 3.5 m s\(^{-1}\) downwards? How far does he descend in this time? [4]

The man has a mass of 80 kg. All resistances to motion may be neglected.

1. Calculate the tension in the wire when the man is being lowered
   1. with an acceleration of 1.5 m s\(^{-2}\) downwards,
   2. with an acceleration of 1.5 m s\(^{-2}\) upwards. [5]

Subsequently, the man is raised and this situation is modelled with a constant resistance of 116 N to his upward motion.

1. For safety reasons, the tension in the wire should not exceed 2500 N. What is the maximum acceleration allowed when the man is being raised? [4]

At another stage of the rescue, the man has equipment of mass 10 kg at the bottom of a vertical rope which is hanging from his waist, as shown in Fig. 6.2. The man and his equipment are being raised; the rope is light and inextensible and the tension in it is 80 N. \includegraphics{figure_6_2}

1. Assuming that the resistance to the upward motion of the man is still 116 N and that there is negligible resistance to the motion of the equipment, calculate the tension in the wire. [4]

A cyclist and her bicycle have a combined mass of 78 kg. While riding on level ground and using her greatest driving force, she is able to accelerate uniformly from rest to 10 ms\(^{-1}\) in 15 seconds against constant resistive forces that total 60 N.

1. Show that her maximum driving force is 112 N. [4 marks]

The cyclist begins to ascend a hill, inclined at an angle \(\alpha\) to the horizontal, riding with her maximum driving force and against the same resistive forces. In this case, she is able to maintain a steady speed.

1. Find the angle \(\alpha\), giving your answer to the nearest degree. [4 marks]
2. Comment on the assumption that the resistive force remains constant
   1. in the case when the cyclist is accelerating,
   2. in the case when she is maintaining a steady speed. [2 marks]

Anila is practising catching tennis balls. She uses a mobile computer-controlled machine which fires tennis balls vertically upwards from a height of 2.5 metres above the ground. Once it has fired a ball, the machine is programmed to move position rapidly to allow Anila time to get into a suitable position to catch the ball. The machine fires a ball at 24 ms\(^{-1}\) vertically upwards and Anila catches the ball just before it touches the ground.

1. Draw a speed-time graph for the motion of the ball from the time it is fired by the machine to the instant before Anila catches it. [3 marks]
2. Find, to the nearest centimetre, the maximum height which the ball reaches above the ground. [4 marks]
3. Calculate the speed at which the ball is travelling when Anila catches it. [4 marks]
4. Calculate the length of time that the ball is in the air. [3 marks]

\includegraphics{figure_3} Figure 3 shows a particle \(X\) of mass 3 kg on a smooth plane inclined at an angle 30° to the horizontal, and a particle \(Y\) of mass 2 kg on a smooth plane inclined at an angle 60° to the horizontal. The two particles are connected by a light, inextensible string of length 2.5 metres passing over a smooth pulley at \(C\) which is the highest point of the two planes. Initially, \(Y\) is at a point just below \(C\) touching the pulley with the string taut. When the particles are released from rest they travel along the lines of greatest slope, \(AC\) in the case of \(X\) and \(BC\) in the case of \(Y\), of their respective planes. \(A\) and \(B\) are the points where the planes meet the horizontal ground and \(AB = 4\) metres.

1. Show that the initial acceleration of the system is given by \(\frac{g}{10}\left(2\sqrt{3} - 3\right)\) ms\(^{-2}\). [7 marks]
2. By finding the tension in the string, or otherwise, find the magnitude of the force exerted on the pulley and the angle that this force makes with the vertical. [7 marks]
3. Find, correct to 2 decimal places, the speed with which \(Y\) hits the ground. [4 marks]

A sports car is being driven along a straight test track. It passes the point \(O\) at time \(t = 0\) at which time it begins to decelerate uniformly. The car passes the points \(L\) and \(M\) at times \(t = 1\) and \(t = 4\) respectively. Given that \(OL\) is 54 m and \(LM\) is 90 m,

1. find the rate of deceleration of the car. [5 marks]

The car subsequently comes to rest at \(N\).

1. Find the distance \(MN\). [4 marks]

A car of mass 1.25 tonnes tows a caravan of mass 0.75 tonnes along a straight, level road. The total resistance to motion experienced by the car and the caravan is 1200 N. The car and caravan accelerate uniformly from rest to 25 m s\(^{-1}\) in 20 seconds.

1. Calculate the driving force produced by the car's engine. [4 marks]

Given that the resistance to motion experienced by the car and by the caravan are in the same ratio as their masses,

1. find these resistances and the tension in the towbar. [4 marks]

When the car and caravan are travelling at a steady speed of 25 m s\(^{-1}\), the towbar snaps. Assuming that the caravan experiences the same resistive force as before,

1. calculate the distance travelled by the caravan before it comes to rest, [5 marks]
2. give a reason why your answer to \((c)\) may be unrealistic. [2 marks]

\includegraphics{figure_1} The points \(A\), \(O\) and \(B\) lie on a straight horizontal track as shown in Figure 1. \(A\) is 20 m from \(O\) and \(B\) is on the other side of \(O\) at a distance \(x\) m from \(O\). At time \(t = 0\), a particle \(P\) starts from rest at \(O\) and moves towards \(B\) with uniform acceleration of 3 m s\(^{-2}\). At the same instant, another particle \(Q\), which is at the point \(A\), is moving with a velocity of 3 m s\(^{-1}\) in the direction of \(O\) with uniform acceleration of 4 m s\(^{-2}\) in the same direction. Given that the \(Q\) collides with \(P\) at \(B\), find the value of \(x\). [10 marks]

Whilst looking over the edge of a vertical cliff, 122.5 metres in height, Jim dislodges a stone. The stone falls freely from rest towards the sea below. Ignoring the effect of air resistance,

1. calculate the time it would take for the stone to reach the sea, [3 marks]
2. find the speed with which the stone would hit the water. [2 marks]

Two seconds after the stone begins to fall, Jim throws a tennis ball downwards at the stone. The tennis ball's initial speed is \(u\) m s\(^{-1}\) and it hits the stone before they both reach the water.

1. Find the minimum value of \(u\). [5 marks]
2. If you had taken air resistance into account in your calculations, what effect would this have had on your answer to part (c)? Explain your answer. [2 marks]

\includegraphics{figure_2} Figure 2 shows two particles \(P\) and \(Q\), of mass 3 kg and 2 kg respectively, attached to the ends of a light, inextensible string which passes over a smooth, fixed pulley. The system is released from rest with \(P\) and \(Q\) at the same level 1.5 metres above the ground and 2 metres below the pulley.

1. Show that the initial acceleration of the system is \(\frac{g}{5}\) m s\(^{-2}\). [4 marks]
2. Find the tension in the string. [2 marks]
3. Find the speed with which \(P\) hits the ground. [3 marks]

When \(P\) hits the ground, it does not rebound.

1. What is the closest that \(Q\) gets to the pulley. [5 marks]

A cyclist is riding up a hill inclined at an angle of 5° to the horizontal. She produces a driving force of 50 N and experiences resistive forces which total 20 N. Given that the combined mass of the cyclist and her bicycle is 70 kg,

1. find, correct to 2 decimal places, the magnitude of the deceleration of the cyclist. [4 marks]

When the cyclist reaches the top of the hill, her speed is 3 m s\(^{-1}\). She subsequently accelerates uniformly so that in the fifth second after she has reached the top of the hill, she travels 12 m.

1. Find her speed at the end of the fifth second. [8 marks]

\includegraphics{figure_2} Figure 2 shows a particle \(A\) of mass 5 kg, lying on a smooth horizontal table which is 0.9 m above the floor. A light inextensible string of length 0.7 m connects \(A\) to a particle \(B\) of mass 2 kg. The string passes over a smooth pulley which is fixed to the edge of the table and \(B\) hangs vertically 0.4 m below the pulley. When the system is released from rest,

1. show that the magnitude of the force exerted on the pulley is \(\frac{10\sqrt{5}}{7}\) g N. [7 marks]
2. find the speed with which \(A\) hits the pulley. [3 marks]

When \(A\) hits the pulley, the string breaks and \(B\) subsequently falls freely under gravity.

1. Find the speed with which \(B\) hits the ground. [4 marks]

Andrew hits a tennis ball vertically upwards towards his sister Barbara who is leaning out of a window 7.5 m above the ground to try to catch it. When the ball leaves Andrew's racket, it is 1.9 m above the ground and travelling at \(21 \text{ m s}^{-1}\). Barbara fails to catch the ball on its way up but succeeds as the ball comes back down. Modelling the ball as a particle and assuming that air resistance can be neglected,

1. find the maximum height above the ground which the ball reaches. [4 marks]
2. find how long Barbara has to wait from the moment that the ball first passes her until she catches it. [6 marks]

\includegraphics{figure_4} Figure 4 shows two golf balls \(P\) and \(Q\) being held at the top of planes inclined at \(30°\) and \(60°\) to the vertical respectively. Both planes slope down to a common hole at \(H\), which is 3 m vertically below \(P\) and \(Q\). \(P\) is released from rest and travels down the line of greatest slope of the plane it is on which is assumed to be smooth.

1. Find the acceleration of \(P\) down the slope. [3 marks]
2. Show that the time taken for \(P\) to reach the hole is 0.904 seconds, correct to 3 significant figures. [5 marks] \(Q\) travels down the line of greatest slope of the plane it is on which is rough. The coefficient of friction between \(Q\) and the plane is \(\mu\). Given that the acceleration of \(Q\) down the slope is \(3 \text{ m s}^{-2}\),
3. find, correct to 3 significant figures, the value of \(\mu\). [5 marks] In order for the two balls to arrive at the hole at the same time, \(Q\) must be released \(t\) seconds before \(P\).
4. Find the value of \(t\) correct to 2 decimal places. [4 marks]

Two cars, P and Q, are being crashed as part of a film 'stunt'. At the start

• P is travelling directly towards Q with a speed of \(8\) ms\(^{-1}\),
• Q is instantaneously at rest and has an acceleration of \(4\) ms\(^{-2}\) directly towards P.

P continues with the same velocity and Q continues with the same acceleration. The cars collide \(T\) seconds after the start.

1. Find expressions in terms of \(T\) for how far each of the cars has travelled since the start. [2]

At the start, P is 90 m from Q.

1. Show that \(T^2 + 4T - 45 = 0\) and hence find \(T\). [5]

In this question take \(g\) as \(10\text{ m s}^{-2}\). A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6. \includegraphics{figure_4} For this model,

1. calculate the distance fallen from \(t = 0\) to \(t = 7\), [3]
2. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction, [3]
3. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\), [3]
4. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). [1]

The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = -\frac{3}{2}t^2 + \frac{19}{2}t + 7\).

1. Verify that \(v\) agrees with the values given in Fig. 6 at \(t = 2\), \(t = 6\) and \(t = 7\). [2]
2. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model. [7]

\includegraphics{figure_6} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards Q of \(0.5\text{ m s}^{-2}\). Particle Q starts 125 m from P at the same time and has a constant speed of \(10\text{ m s}^{-1}\) away from P. The initial values are shown in Fig. 4.

1. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion. [2]
2. How much time does it take for P to catch up with Q and how far does P travel in this time? [5]