Multi-stage motion with all parameters given

8 A tram travels from stop \(A\) to stop \(B\), a distance of 300 m . First the tram starts from rest at \(A\) and accelerates uniformly at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 16 seconds. Then it travels at a constant speed and finally it slows down uniformly at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) coming to rest at \(B\).

1. Sketch the velocity-time graph for the journey of the tram from \(A\) to \(B\).
2. Find the speed of the tram and the distance travelled at the end of the first 16 seconds.
3. Show that the journey from \(A\) to \(B\) takes 49.5 seconds.

An aircraft moves along a straight horizontal runway with a constant acceleration of \(1.5 \mathrm {~ms} ^ { - 2 }\). Points \(A\) and \(B\) lie on the runway. The aircraft passes \(A\) with speed \(4 \mathrm {~ms} ^ { - 1 }\) and its speed at \(B\) must be at least \(78 \mathrm {~ms} ^ { - 1 }\) if it is to take off successfully. a) Find the speed of the aircraft 8 seconds after it passes \(A\).
b) Determine the minimum value of the distance \(A B\) for the aircraft to take off successfully. The diagram below shows an object \(A\), of mass 15 kg , lying on a smooth horizontal surface. It is connected to a box \(B\) by a light inextensible string which passes over a smooth pulley \(P\), fixed at the edge of the surface, so that box \(B\) hangs freely. An object \(C\) lies on the horizontal floor of box \(B\) so that the combined mass of \(B\) and \(C\) is 10 kg . \includegraphics[max width=\textwidth, alt={}, center]{77c62e6d-58e4-42d3-9982-5a8325e8e826-09_661_862_614_598} Initially, the system is held at rest with the string just taut. A horizontal force of magnitude 150 N is then applied to \(A\) in the direction \(P A\) so that box \(B\) is raised.
a) Find the magnitude of the acceleration of \(A\) and the tension in the string.
b) Given that object \(C\) has mass 4 kg , calculate the reaction of the floor of the box on object \(C\).
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1 In this question, \(\mathbf { i }\) and \(\mathbf { j }\) represent unit vectors due east and due north respectively. Sarah is going for a walk. She leaves her house and walks directly to the shop. She then walks directly from the shop to the park. Relative to her house:

• the shop has position vector \(\left( - \frac { 2 } { 3 } \mathbf { j } \right) \mathrm { km }\),
• the park is 2 km away on a bearing of \(060 ^ { \circ }\).
  a) Show that the position vector of the park relative to the house is \(( \sqrt { 3 } \mathbf { i } + \mathbf { j } ) \mathrm { km }\).
  b) Determine the total distance walked by Sarah from her house to the park.
  c) By considering a modelling assumption you have made, explain why the answer you found in part (b) may not be the actual distance that Sarah walked.

Two cyclists, Isabella and Maria, are having a race. They both travel along a straight road with constant acceleration, starting from rest at point \(A\). Isabella accelerates for 5 s at a constant rate \(a \text{ m s}^{-2}\). She then travels at the constant speed she has reached for 10 s, before decelerating to rest at a constant rate over a period of 5 s. Maria accelerates at a constant rate, reaching a speed of \(5 \text{ m s}^{-1}\) in a distance of 27.5 m. She then maintains this speed for a period of 10 s, before decelerating to rest at a constant rate over a period of 5 s.

1. Given that \(a = 1.1\), find which cyclist travels further. [5]
2. Find the value of \(a\) for which the two cyclists travel the same distance. [2]

A car starts from rest and moves in a straight line with constant acceleration for a distance of 200 m, reaching a speed of 25 m s\(^{-1}\). The car then travels at this speed for 400 m, before decelerating uniformly to rest over a period of 5 s.

1. Find the time for which the car is accelerating. [2]
2. Sketch the velocity–time graph for the motion of the car, showing the key points. [2]
3. Find the average speed of the car during its motion. [2]

A train starts from rest at a station \(A\) and moves along a straight horizontal track. For the first 10 s, the train moves with constant acceleration 1.2 m s\(^{-2}\). For the next 24 s it moves at a constant acceleration 0.75 m s\(^{-2}\). It then moves with constant speed for \(T\) seconds. Finally it slows down with constant deceleration 3 m s\(^{-2}\) until it comes to a rest at station \(B\).

1. Show that, 34 s after leaving \(A\), the speed of the train is 30 m s\(^{-1}\). [3]
2. Sketch a speed-time graph to illustrate the motion of the train as it moves from \(A\) to \(B\). [3]
3. Find the distance moved by the train during the first 34 s of its journey from \(A\). [4]

The distance from \(A\) to \(B\) is 3 km.

1. Find the value of \(T\). [4]

A man travels in a lift to the top of a tall office block. The lift starts from rest on the ground floor and moves vertically. It comes to rest again at the top floor, having moved a vertical distance of \(27 \text{ m}\). The lift initially accelerates with a constant acceleration of \(2 \text{ m s}^{-1}\) until it reaches a speed of \(3 \text{ m s}^{-1}\). It then moves with a constant speed of \(3 \text{ m s}^{-1}\) for \(T\) seconds. Finally it decelerates with a constant deceleration for \(2.5 \text{ s}\) before coming to rest at the top floor.

1. Sketch a speed-time graph for the motion of the lift. [2]
2. Hence, or otherwise, find the value of \(T\). [3]
3. Sketch an acceleration-time graph for the motion of the lift. [3]

The mass of the man is \(80 \text{ kg}\) and the mass of the lift is \(120 \text{ kg}\). The lift is pulled up by means of a vertical cable attached to the top of the lift. By modelling the cable as light and inextensible, find

1. the tension in the cable when the lift is accelerating, [3]
2. the magnitude of the force exerted by the lift on the man during the last \(2.5 \text{ s}\) of the motion. [3]

A train is travelling at \(10 \text{ m s}^{-1}\) on a straight horizontal track. The driver sees a red signal 135 m ahead and immediately applies the brakes. The train immediately decelerates with constant deceleration for 12 s, reducing its speed to \(3 \text{ m s}^{-1}\). The driver then releases the brakes and allows the train to travel at a constant speed of \(3 \text{ m s}^{-1}\) for a further 15 s. He then applies the brakes again and the train slows down with constant deceleration, coming to rest as it reaches the signal.

1. Sketch a speed-time graph to show the motion of the train, [3]
2. Find the distance travelled by the train from the moment when the brakes are first applied to the moment when its speed first reaches \(3 \text{ m s}^{-1}\). [2]
3. Find the total time from the moment when the brakes are first applied to the moment when the train comes to rest. [5]

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\(^{-1}\).

1. Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race. [2]
2. Find the distance run by the girl in the first 64 s of the race. [3]
3. Find the value of \(V\). [5]
4. Find the deceleration of the girl in the final 20 s of her race. [2]

A car starts from rest at a point \(O\) and moves in a straight line. The car moves with constant acceleration \(4 \text{ m s}^{-2}\) until it passes the point \(A\) when it is moving with speed \(10 \text{ m s}^{-1}\). It then moves with constant acceleration \(3 \text{ m s}^{-2}\) for 6 s until it reaches the point \(B\). Find

1. the speed of the car at \(B\), [2]
2. the distance \(OB\). [5]

A train \(T_1\) moves from rest at Station \(A\) with constant acceleration \(2 \text{ m s}^{-2}\) until it reaches a speed of \(36 \text{ m s}^{-1}\). In maintains this constant speed for 90 s before the brakes are applied, which produce constant retardation \(3 \text{ m s}^{-2}\). The train \(T_1\) comes to rest at station \(B\).

1. Sketch a speed-time graph to illustrate the journey of \(T_1\) from \(A\) to \(B\). [3]
2. Show that the distance between \(A\) and \(B\) is 3780 m. [5]

\includegraphics{figure_3} A second train \(T_2\) takes 150 s to move form rest at \(A\) to rest at \(B\). Figure 3 shows the speed-time graph illustrating this journey.

1. Explain briefly one way in which \(T_1\)'s journey differs from \(T_2\)'s journey. [1]
2. Find the greatest speed, in m s\(^{-1}\), attained by \(T_2\) during its journey. [3]

A car starts from rest at time \(t = 0\) and moves along a straight road with constant acceleration 4 ms\(^{-2}\) for 10 seconds. It then travels at a constant speed for 50 seconds before decelerating to rest over a further distance of 240 m.

1. Sketch a graph of velocity against time for the total period of the car's motion. [3 marks]
2. Find the car's average speed for the whole journey. [6 marks]

In reality the car's acceleration \(a\) ms\(^{-2}\) in the first 10 seconds is not constant, but increases from 0 to 4 ms\(^{-2}\) in the first 5 seconds and then decreases to 0 again. A refined model designed to take account of this uses the formula \(a = k(mt - t^2)\) for \(0 \leq t \leq 10\).

1. Calculate the values of the constants \(k\) and \(m\). [5 marks]
2. Find the acceleration of the car when \(t = 2\) according to this model. [2 marks]

A vehicle moves along a straight horizontal road. Points \(A\) and \(B\) lie on the road. As the vehicle passes point \(A\), it is moving with constant speed 15 ms\(^{-1}\). It travels with this constant speed for 2 minutes before a constant deceleration is applied for 12 seconds so that it comes to rest at point \(B\).

1. Find the distance \(AB\). [3]

The vehicle then reverses with a constant acceleration of 2 ms\(^{-2}\) for 8 seconds, followed by a constant deceleration of 1·6 ms\(^{-2}\), coming to rest at the point \(C\), which is between \(A\) and \(B\).

1. Calculate the time it takes for the vehicle to reverse from \(B\) to \(C\). [4]
2. Sketch a velocity-time graph for the motion of the vehicle. [3]
3. Determine the distance \(AC\). [2]