Average speed or total distance calculation

5 A small box B of weight 400 N is held in equilibrium by two light strings AB and BC . The string BC is fixed at C . The end A of string AB is fixed so that AB is at an angle \(\alpha\) to the vertical where \(\alpha < 60 ^ { \circ }\). String BC is at \(60 ^ { \circ }\) to the vertical. This information is shown in Fig. 5. \begin{figure}[h] \end{figure}

1. Draw a labelled diagram showing all the forces acting on the box.
2. In one situation string AB is fixed so that \(\alpha = 30 ^ { \circ }\). By drawing a triangle of forces, or otherwise, calculate the tension in the string BC and the tension in the string AB .
3. Show carefully, but briefly, that the box cannot be in equilibrium if \(\alpha = 60 ^ { \circ }\) and BC remains at \(60 ^ { \circ }\) to the vertical. 7 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9a79f274-1a3f-4d11-9775-313d82075035-004_341_1107_484_498} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
4. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
5. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
6. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
7. Calculate the direction of the motion of the stone at \(\mathbf { C }\). Section B (36 marks)

11 \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-08_586_672_1231_242} A particle \(P\) moves along the \(x\)-axis. At time \(t\) seconds, where \(t \geqslant 0\), the velocity of \(P\) in the positive \(x\)-direction is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It is given that \(v = t ( t - 3 ) ( 8 - t )\). \(P\) attains its maximum velocity at time \(T\) seconds. The diagram shows part of the velocity-time graph for the motion of \(P\).

1. State the acceleration of \(P\) at time \(T\).
2. In this question you must show detailed reasoning. Determine the value of \(T\).
3. Find the total distance that \(P\) travels between times \(t = 0\) and \(t = T\). \includegraphics[max width=\textwidth, alt={}, center]{a1f4ccbd-f5ed-437a-ae76-c4925ce86e25-09_524_410_251_242} Particles \(P\) and \(Q\), of masses 4 kg and 6 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is in equilibrium with \(P\) hanging 1.75 m above a horizontal plane and \(Q\) resting on the plane. Both parts of the string below the pulley are vertical (see diagram).
   1. Find the magnitude of the normal reaction force acting on \(Q\). The mass of \(P\) is doubled, and the system is released from rest. You may assume that in the subsequent motion \(Q\) does not reach the pulley.
   2. Determine the magnitude of the force exerted on the pulley by the string before \(P\) strikes the plane.
   3. Determine the total distance travelled by \(Q\) between the instant when the system is released and the instant when \(Q\) first comes momentarily to rest. When this motion is observed in practice, it is found that the total distance travelled by \(Q\) between the instant when the system is released and the instant when \(Q\) first comes momentarily to rest is less than the answer calculated in part (c).
   4. State one factor that could account for this difference.

7 A toy boat of mass 1.5 kg is pushed across a pond, starting from rest, for 2.5 seconds. During this time, the boat has an acceleration of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Subsequently, when the only horizontal force acting on the boat is a constant resistance to motion, the boat travels 10 m before coming to rest. Calculate the magnitude of the resistance to motion.

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.

3 A pair of cameras records the time that it takes a car on a motorway to travel a distance of 2000 metres. A car passes the first camera whilst travelling at \(32 \mathrm {~ms} ^ { - 1 }\). The car continues at this speed for 12.5 seconds and then decelerates uniformly until it passes the second camera when its speed has decreased to \(18 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the distance travelled by the car in the first 12.5 seconds.
2. Find the time for which the car is decelerating.
3. Sketch a speed-time graph for the car on this 2000-metre stretch of motorway.
4. Find the average speed of the car on this 2000-metre stretch of motorway.

1. \begin{figure}[h] \end{figure} Figure 1 shows a distance-time graph for a car journey from Birmingham to Newquay which included a stop for lunch at a service station near Exeter. During the first part of the journey three-quarters of the total distance, \(d\), was covered in 3 hours. After a 1 hour stop, the remaining distance was completed in 2 hours.

1. Calculate, in the form \(k : 1\), the ratio of the average speed during the first 3 hours of the journey to the average speed during the last 2 hours of the journey.
   (4 marks)
   Given that the average speed of the car over the whole journey (excluding the stop) was \(80 \mathrm { kmh } ^ { - 1 }\),
2. find the average speed of the car on the first part of the journey.
   (4 marks)

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.