3.02c Interpret kinematic graphs: gradient and area

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. 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}

6 In this question take \(g\) as \(10 \mathrm {~m \mathrm {~s} ^ { - 2 }\).} 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. \begin{figure}[h] \end{figure} For this model,

1. calculate the distance fallen from \(t = 0\) to \(t = 7\),
2. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction,
3. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\),
4. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = - \frac { 3 } { 2 } t ^ { 2 } + \frac { 19 } { 2 } t + 7\).
5. Verify that \(v\) agrees with the values given in Fig. 6 at \(t = 2 , t = 6\) and \(t = 7\).
6. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model.

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.

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.

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).

13 A vehicle, of total mass 1200 kg , is travelling along a straight, horizontal road at a constant speed of \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) This vehicle begins to accelerate at a constant rate.
After 40 metres it reaches a speed of \(17 \mathrm {~ms} ^ { - 1 }\) Find the resultant force acting on the vehicle during the period of acceleration.

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. 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.

6. \begin{figure}[h] \end{figure} A car moves along a straight horizontal road. At time \(t = 0\), the velocity of the car is \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then accelerates with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds. The car travels a distance \(D\) metres during these \(T\) seconds. Figure 1 shows the velocity-time graph for the motion of the car for \(0 \leqslant t \leqslant T\).
Using the graph, show that \(D = U T + 1 / 2 a T ^ { 2 }\).
(No credit will be given for answers which use any of the kinematics (suvat) formulae listed under Mechanics in the AS Mathematics section of the formulae booklet.)

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 \(t = 2\).

A particle \(P\) moves along the \(x\)-axis so that its velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds \(( t \geqslant 0 )\) is given by $$v = 3 t ^ { 2 } - 24 t + 36$$ a) Find the values of \(t\) when \(P\) is instantaneously at rest.
b) Calculate the total distance travelled by the particle \(P\) whilst its velocity is decreasing.

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 particle \(P\) moves in a straight line. The velocity \(v\text{ms}^{-1}\) at time \(t\) seconds is given by $$v = 0.5t \quad \text{for } 0 \leqslant t \leqslant 10,$$ $$v = 0.25t^2 - 8t + 60 \quad \text{for } 10 < t \leqslant 20.$$

1. Show that there is an instantaneous change in the acceleration of the particle at \(t = 10\). [3]
2. Find the total distance covered by \(P\) in the interval \(0 \leqslant t \leqslant 20\). [6]

\includegraphics{figure_4} The velocity of a particle at time \(t\) s after leaving a fixed point \(O\) is \(v\) m s\(^{-1}\). The diagram shows a velocity-time graph which models the motion of the particle. The graph consists of \(5\) straight line segments. The particle accelerates to a speed of \(0.9\) m s\(^{-1}\) in a period of \(3\) s, then travels at constant speed for \(6\) s, then comes instantaneously to rest \(1\) s later. The particle then moves back and returns to rest at \(O\) at time \(T\) s.

1. Find the distance travelled by the particle in the first \(10\) s of its motion. [2]
2. Given that \(T = 12\), find the minimum velocity of the particle. [2]
3. Given instead that the greatest speed of the particle is \(3\) m s\(^{-1}\), find the value of \(T\) and hence find the average speed of the particle for the whole of the motion. [4]

\includegraphics{figure_1} The displacement of a particle at time \(t\) s after leaving a fixed point \(O\) is \(s\) m. The diagram shows a displacement-time graph which models the motion of the particle. The graph consists of 4 straight line segments. The particle travels 50 m in the first 10 s, then travels at \(2\) m s\(^{-1}\) for a period of 10 s. The particle then comes to rest for a period of 20 s, before returning to its starting point when \(t = 60\).

1. Find the velocity of the particle during the last 20 s of its motion. [2]
2. Sketch a velocity-time graph for the motion of the particle from \(t = 0\) to \(t = 60\). [3]

\includegraphics{figure_6} The diagram shows the velocity-time graph for a lift moving between floors in a building. The graph consists of straight line segments. In the first stage the lift travels downwards from the ground floor for \(5 \text{ s}\), coming to rest at the basement after travelling \(10 \text{ m}\).

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

The second stage consists of a \(10 \text{ s}\) wait at the basement. In the third stage, the lift travels upwards until it comes to rest at a floor \(34.5 \text{ m}\) above the basement, arriving \(24.5 \text{ s}\) after the start of the first stage. The lift accelerates at \(2 \text{ m s}^{-2}\) for the first \(3 \text{ s}\) of the third stage, reaching a speed of \(V \text{ m s}^{-1}\). Find

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

\includegraphics{figure_2} The diagram shows the velocity-time graph for the motion of a machine's cutting tool. The graph consists of five straight line segments. The tool moves forward for 8 s while cutting and then takes 3 s to return to its starting position. Find

1. the acceleration of the tool during the first 2 s of the motion, [1]
2. the distance the tool moves forward while cutting, [2]
3. the greatest speed of the tool during the return to its starting position. [2]