3.02b Kinematic graphs: displacement-time and velocity-time

1 A ring is moving up and down a vertical pole. The displacement, \(s \mathrm {~m}\), of the ring above a mark on the pole is modelled by the displacement-time graph shown in Fig. 1. The three sections of the graph are straight lines. \begin{figure}[h] \end{figure}

1. Calculate the velocity of the ring in the interval \(0 < t < 2\) and in the interval \(2 < t < 3.5\).
2. Sketch a velocity-time graph for the motion of the ring during the 4 seconds.
3. State the direction of motion of the ring when
   (A) \(t = 1\),
   (B) \(t = 2.75\),
   (C) \(t = 3.25\).

2 Fig. 2 shows an acceleration-time graph modelling the motion of a particle. \begin{figure}[h] \end{figure} At \(t = 0\) the particle has a velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive direction.

1. Find the velocity of the particle when \(t = 2\).
2. At what time is the particle travelling in the negative direction with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) ?

3 A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.

1. Sketch a velocity-time graph for the motion.
2. Calculate the distance travelled by the cyclist.

4 Fig. 1 is the velocity-time graph for the motion of a body. The velocity of the body is \(v \mathrm {~m} \mathrm {~s} { } ^ { 1 }\) at time \(t\) seconds. \begin{figure}[h] \end{figure} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m . Find the value of \(V\).

3 Fig. 1 shows the speed-time graph of a runner during part of his training. \begin{figure}[h] \end{figure} For each of the following statements, say whether it is true or false. If it is false give a brief explanation.
(A) The graph shows that the runner finishes where he started.
(B) The runner's maximum speed is \(8 \mathrm {~ms} ^ { - 1 }\).
(C) At time 58 seconds, the runner is slowing down at a rate of \(1.6 \mathrm {~ms} ^ { - 2 }\).
(D) The runner travels 400 m altogether.

1. \hspace{0pt} [In this question the horizontal unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are due east and due north respectively.]

A ship \(A\) has constant velocity \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { kmh } ^ { - 1 }\) and a ship \(B\) has constant velocity \(( - \mathbf { i } + 3 \mathbf { j } ) \mathrm { km } \mathrm { h } ^ { - 1 }\). At noon, the position vectors of the ships \(A\) and \(B\) with respect to a fixed origin \(O\) are \(( - 2 \mathbf { i } + \mathbf { j } ) \mathrm { km }\) and \(( 5 \mathbf { i } - 2 \mathbf { j } ) \mathrm { km }\) respectively. Find

1. the time at which the two ships are closest together,
2. the length of time for which ship \(A\) is within 2 km of ship \(B\).

5. A horizontal square field, \(P Q R S\), has sides of length 75 m . Ali is at corner \(P\) of the field and Beth is at corner \(Q\) of the field. Ali starts to walk in a straight line along the diagonal of the field from \(P\) to \(R\) at a constant speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Beth sees Ali start to walk, waits 10 seconds, and then walks from \(Q\) to intercept Ali. Beth walks in a straight line at a constant speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the time from the instant Beth leaves \(Q\) until the instant that she intercepts Ali,
2. the direction Beth should take.

2 A particle, Q , moves so that its velocity, \(\mathbf { v }\), at time \(t\) is given by \(\mathbf { v } = ( 6 t - 6 ) \mathbf { i } + \left( 3 - 2 t + t ^ { 2 } \right) \mathbf { j } + 4 \mathbf { k }\), where \(0 \leqslant t \leqslant 6\).

1. Explain how you know that Q is never stationary. When Q is at a point A the direction of the acceleration of Q is parallel to the \(\mathbf { i }\) direction. When Q is at a point B the direction of the acceleration of Q makes an angle of \(45 ^ { \circ }\) with the \(\mathbf { i }\) direction.
2. Determine the straight-line distance AB .

1 A particle, P , has velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds given by \(\mathbf { v } = \left( \begin{array} { c } 6 \left( t ^ { 2 } - 3 t + 2 \right) \\ 2 ( 1 - t ) \\ 3 \left( t ^ { 2 } - 1 \right) \end{array} \right)\), where \(0 \leq t \leq 3\).

1. Show that there is just one time at which P is instantaneously at rest and state this value of \(t\). P has a mass of 5 kg and is acted on by a single force \(\mathbf { F }\) N.
2. Find \(\mathbf { F }\) when \(t = 2\).
3. Find an expression for the position, \(\mathbf { r } \mathrm { m }\), of P at time \(t \mathrm {~s}\), given that \(\mathbf { r } = \left( \begin{array} { c } - 5 \\ 2 \\ 6 \end{array} \right)\) when \(t = 0\).

1. A particle \(P\) moves on the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and moving with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction of \(x\) increasing, where

$$v = 5 \sin 2 t$$ When \(t = 0 , x = 1\) and \(P\) is at rest.

1. Find the magnitude and direction of the acceleration of \(P\) at the instant when \(P\) is next at rest.
2. Show that \(1 \leqslant x \leqslant 6\)
3. Find the total time, in the first \(4 \pi\) seconds of the motion, for which \(P\) is more than 3 metres from \(O\)
   \includegraphics[max width=\textwidth, alt={}]{a7901165-1679-4d30-9444-0c27020e32ea-16_2260_52_309_1982}

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.

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).

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).

14 A motorised scooter is travelling along a straight path with velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over time \(t\) seconds as shown by the following graph. \includegraphics[max width=\textwidth, alt={}, center]{b7df05bf-f3fc-4705-a13c-6b562896fa9f-20_1120_1134_420_452} Noosha says that, in the period \(\mathbf { 1 2 } \leq \boldsymbol { t } \leq \mathbf { 3 6 }\), the scooter travels approximately 130 metres. Determine if Noosha is correct, showing clearly any calculations you have used.

15 A car is moving in a straight line along a horizontal road. The graph below shows how the car's velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) changes with time, \(t\) seconds. \includegraphics[max width=\textwidth, alt={}, center]{ad6590e8-6673-45ca-bef3-a14716978827-23_509_746_456_648} Over the period \(0 \leq t \leq 15\) the car has a total displacement of - 7 metres.
Initially the car has velocity \(0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the next time when the velocity of the car is \(0 \mathrm {~ms} ^ { - 1 }\) [0pt] [4 marks]

12 A particle moves in a straight line.
After the first 4 seconds of its motion, the displacement of the particle from its initial position is 0 metres. One of the graphs on the opposite page shows the velocity \(v \mathrm {~ms} ^ { - 1 }\) of the particle after time \(t\) seconds of its motion. Identify the correct graph.
Tick ( \(\checkmark\) ) one box. \includegraphics[max width=\textwidth, alt={}, center]{de8a7d38-a665-4feb-854e-ac83f413d133-19_2249_896_260_484}

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}

6 A particle travels along a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) after \(t\) seconds is given by $$v = t ^ { 3 } - 6 t ^ { 2 } + 8 t \text { for } 0 \leqslant t \leqslant 4$$ When \(t = 0\) the particle is at rest at the point \(P\).

1. Find the times (other than \(t = 0\) ) when the particle is at rest. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\).
2. Find the acceleration of the particle when \(t = 2\).
3. Find an expression for the displacement of the particle from \(P\) after \(t\) seconds. Hence state its displacement from \(P\) when \(t = 2\) and find its average speed between \(t = 0\) and \(t = 2\).

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.

11 In a training exercise, a submarine is travelling due north at \(15 \mathrm {~km} \mathrm {~h} ^ { - 1 }\). The submarine commander sees his target 5 km away on a bearing of \(310 ^ { \circ }\). The target is travelling due east at \(20 \mathrm {~km} \mathrm {~h} ^ { - 1 }\).

1. If each of the submarine and target maintains its present course and speed, find the shortest distance between them.
2. In fact, as soon as he sees the target, the submarine commander changes course, without changing speed, so as to intercept the target as quickly as possible. Find
   1. the course, in degrees, set by the submarine commander,
   2. the time taken, in minutes, to intercept the target from the moment that the course changes.

6 A crate, which has a mass of 220 kg , is being lowered on the end of a cable onto the back of a lorry.

1. Draw a diagram to show the forces acting on the crate. The crate is lowered in three stages.
   Stage 1 It starts from rest and accelerates at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Stage 2 It descends at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Stage 3 It decelerates at \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and eventually comes to rest.
2. Find the tension in the cable in each of the three stages.
3. Sketch the velocity-time graph for the complete downward motion of the crate.
4. The crate is lowered 15 m altogether. By considering your velocity-time graph, find the total time taken.

A tram starts from rest and moves with uniform acceleration for 20 s. The tram then travels at a constant speed, \(V \text{ ms}^{-1}\), for 170 s before being brought to rest with a uniform deceleration of magnitude twice that of the acceleration. The total distance travelled by the tram is 2.775 km.

1. Sketch a velocity-time graph for the motion, stating the total time for which the tram is moving. [2]
2. Find \(V\). [2]
3. Find the magnitude of the acceleration. [2]

A particle \(P\) moves in a straight line. The velocity \(v \text{ ms}^{-1}\) at time \(t\) s is given by $$v = 2t + 1 \quad \text{for } 0 \leqslant t \leqslant 5,$$ $$v = 36 - t^2 \quad \text{for } 5 \leqslant t \leqslant 7,$$ $$v = 2t - 27 \quad \text{for } 7 \leqslant t \leqslant 13.5.$$

1. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 13.5\). [3]
2. Find the acceleration at the instant when \(t = 6\). [2]
3. Find the total distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 13.5\). [5]

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]