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

3. \begin{figure}[h] \end{figure} A car of mass 1200 kg moves along a straight horizontal road. In order to obey a speed restriction, the brakes of the car are applied for 3 s , reducing the car's speed from \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The brakes are then released and the car continues at a constant speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for a further 4 s . Figure 2 shows a sketch of the speed-time graph of the car during the 7 s interval. The graph consists of two straight line segments.

1. Find the total distance moved by the car during this 7 s interval.
2. Explain briefly how the speed-time graph shows that, when the brakes are applied, the car experiences a constant retarding force.
3. Find the magnitude of this retarding force.

7. Two trains \(A\) and \(B\) run on parallel straight tracks. Initially both are at rest in a station and level with each other. At time \(t = 0 , A\) starts to move. It moves with constant acceleration for 12 s up to a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and then moves at a constant speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) starts to move in the same direction as \(A\) when \(t = 40\), where \(t\) is measured in seconds. It accelerates with the same initial acceleration as \(A\), up to a speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then moves at a constant speed of \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Train \(B\) overtakes \(A\) after both trains have reached their maximum speed. Train \(B\) overtakes \(A\) when \(t = T\).

1. Sketch, on the same diagram, the speed-time graphs of both trains for \(0 \leq t \leq T\).
2. Find the value of \(T\).

1. Figure 1 \includegraphics[max width=\textwidth, alt={}, center]{3a8395fd-6e44-48a1-8c97-3365a284956a-02_404_755_312_577} Figure 1 shows the speed-time graph of a cyclist moving on a straight road over a 7 s period. The sections of the graph from \(t = 0\) to \(t = 3\), and from \(t = 3\) to \(t = 7\), are straight lines. The section from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis. State what can be deduced about the motion of the cyclist from the fact that

1. the graph from \(t = 0\) to \(t = 3\) is a straight line,
2. the graph from \(t = 3\) to \(t = 7\) is parallel to the \(t\)-axis.
3. Find the distance travelled by the cyclist during this 7 s period.

4. A car is moving along a straight horizontal road. The speed of the car as it passes the point \(A\) is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the car maintains this speed for 30 s . The car then decelerates uniformly to a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is then maintained until the car passes the point \(B\). The time taken to travel from \(A\) to \(B\) is 90 s and \(A B = 1410 \mathrm {~m}\).

1. Sketch, in the space below, a speed-time graph to show the motion of the car from \(A\) to \(B\).
2. Calculate the deceleration of the car as it decelerates from \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Question 4 continued \(\_\_\_\_\)

3. A car starts from rest and moves with constant acceleration along a straight horizontal road. The car reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 20 seconds. It moves at constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 30 seconds, then moves with constant deceleration \(\frac { 1 } { 2 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it has speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It moves at speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next 15 seconds and then moves with constant deceleration \(\frac { 1 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest.

1. Sketch, in the space below, a speed-time graph for this journey. In the first 20 seconds of this journey the car travels 140 m . Find
2. the value of \(V\),
3. the total time for this journey,
4. the total distance travelled by the car.

7. A train travels along a straight horizontal track between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(V \mathrm {~ms} ^ { - 1 } , ( V < 50 )\). The train then travels at this constant speed before it moves with constant deceleration \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at \(B\).

1. Sketch in the space below a speed-time graph for the motion of the train between the two stations \(A\) and \(B\). The total time for the journey from \(A\) to \(B\) is 5 minutes.
2. Find, in terms of \(V\), the length of time, in seconds, for which the train is
   1. accelerating,
   2. decelerating,
   3. moving with constant speed. Given that the distance between the two stations \(A\) and \(B\) is 6.3 km ,
3. find the value of \(V\).

4. Two trains \(M\) and \(N\) are moving in the same direction along parallel straight horizontal tracks. At time \(t = 0 , M\) overtakes \(N\) whilst they are travelling with speeds \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. Train \(M\) overtakes train \(N\) as they pass a point \(X\) at the side of the tracks. After overtaking \(N\), train \(M\) maintains its speed of \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T\) seconds and then decelerates uniformly, coming to rest next to a point \(Y\) at the side of the tracks. After being overtaken, train \(N\) maintains its speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 25 s and then decelerates uniformly, also coming to rest next to the point \(Y\). The times taken by the trains to travel between \(X\) and \(Y\) are the same.

1. Sketch, on the same diagram, the speed-time graphs for the motions of the two trains between \(X\) and \(Y\). Given that \(X Y = 975 \mathrm {~m}\),
2. find the value of \(T\).
   

3. A car accelerates uniformly from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in \(T\) seconds. The car then travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(4 T\) seconds and finally decelerates uniformly to rest in a further 50 s .

1. Sketch a speed-time graph to show the motion of the car. The total distance travelled by the car is 1220 m . Find
2. the value of \(T\),
3. the initial acceleration of the car.

5. A particle moves along the \(x\)-axis. At time \(t\) seconds, \(t \geqslant 0\), the velocity of the particle is \(v \mathrm {~ms} ^ { - 1 }\) in the direction of \(x\) increasing, where \(v = 2 t ^ { \frac { 3 } { 2 } } - 6 t + 2\) At time \(t = 0\) the particle passes through the origin \(O\). At the instant when the acceleration of the particle is zero, the particle is at the point \(A\). Find the distance \(O A\).
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8. Two trains, \(A\) and \(B\), start together from rest, at time \(t = 0\), at a station and move along parallel straight horizontal tracks. Both trains come to rest at the next station after 180 s . Train \(A\) moves with constant acceleration \(\frac { 2 } { 3 } \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 30 s , then moves at constant speed for 120 s and then moves with constant deceleration for the final 30 s . Train \(B\) moves with constant acceleration for 90 s and then moves with constant deceleration for the final 90 s .

1. Sketch, on the same axes, the speed-time graphs for the motion of the two trains between the two stations.
2. Find the acceleration of train \(B\) for the first half of its journey.
3. Find the times when the two trains are moving at the same speed.
4. Find the distance between the trains 96 s after they start. \includegraphics[max width=\textwidth, alt={}, center]{fade35da-8dca-4d98-a07c-ed3a173fccda-28_43_58_2457_1893}

5. A cyclist is travelling along a straight horizontal road. The cyclist starts from rest at point \(A\) on the road and accelerates uniformly at \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 20 seconds. He then moves at constant speed for \(4 T\) seconds, where \(T < 20\). He then decelerates uniformly at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and after \(T\) seconds passes through point \(B\) on the road. The distance from \(A\) to \(B\) is 705 m .

1. Sketch a speed-time graph for the motion of the cyclist between points \(A\) and \(B\).
2. Find the value of \(T\). The cyclist continues his journey, still decelerating uniformly at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), until he comes to rest at point \(C\) on the road.
3. Find the total time taken by the cyclist to travel from \(A\) to \(C\).

6. A train travels for a total of 270 s along a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration for 60 s until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train then travels at this constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before it moves with constant deceleration for 30 s , coming to rest at \(B\).

1. Sketch below a speed-time graph for the journey of the train between the two stations \(A\) and \(B\). Given that the distance between the two stations is 4.5 km ,
2. find the value of \(V\),
3. find how long it takes the train to travel from station \(A\) to the point that is exactly halfway between the two stations. The train is travelling at speed \(\frac { 1 } { 4 } V \mathrm {~ms} ^ { - 1 }\) at times \(T _ { 1 }\) seconds and \(T _ { 2 }\) seconds after leaving station \(A\).
4. Find the value of \(T _ { 1 }\) and the value of \(T _ { 2 }\)

5. A car travels at a constant speed of \(40 \mathrm {~ms} ^ { - 1 }\) in a straight line along a horizontal racetrack. At time \(t = 0\), the car passes a motorcyclist who is at rest. The motorcyclist immediately sets off to catch up with the car. The motorcyclist accelerates at \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 15 s and then accelerates at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a further \(T\) seconds until he catches up with the car.

1. Sketch, on the same axes, the speed-time graph for the motion of the car and the speed-time graph for the motion of the motorcyclist, from time \(t = 0\) to the instant when the motorcyclist catches up with the car. At the instant when \(t = t _ { 1 }\) seconds, the car and the motorcyclist are moving at the same speed.
2. Find the value of \(t _ { 1 }\)
3. Show that \(T ^ { 2 } + k T - 300 = 0\), where \(k\) is a constant to be found. DO NOT WRITEIN THIS AREA
   

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7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps out of the helicopter and immediately falls vertically and freely under gravity from rest for 2.5 s . His parachute then opens and causes him to immediately decelerate at a constant rate of \(3.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for \(T\) seconds ( \(T < 6\) ), until his speed is reduced to \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He then moves with this constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until he hits the ground. While he is decelerating, he falls a distance of 73.75 m . The total time between the instant when he leaves \(H\) and the instant when he hits the ground is 20 s . The parachutist is modelled as a particle.

1. Find the speed of the parachutist at the instant when his parachute opens.
2. Sketch a speed-time graph for the motion of the parachutist from the instant when he leaves \(H\) to the instant when he hits the ground.
3. Find the value of \(T\).
4. Find, to the nearest metre, the height of the point \(H\) above the ground.
   7. A helicopter is hovering at rest above horizontal ground at the point \(H\). A parachutist steps

4. At time \(t = 0\), a small ball is projected vertically upwards from a point \(A\) which is 24.5 m above the ground. The ball first comes to instantaneous rest at the point \(B\), where \(A B = 19.6 \mathrm {~m}\) and first hits the ground at time \(t = T\) seconds. The ball is modelled as a particle moving freely under gravity.

1. Find the value of \(T\).
2. Sketch a speed-time graph for the motion of the ball from \(t = 0\) to \(t = T\) seconds.
   (No further calculations are needed in order to draw this sketch.)

1. A train travels along a straight horizontal track between two stations \(A\) and \(B\).

The train starts from rest at station \(A\) and accelerates uniformly for \(T\) seconds until it reaches a speed of \(20 \mathrm {~ms} ^ { - 1 }\) The train then travels at a constant speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 3 minutes before decelerating uniformly until it comes to rest at station \(B\). The magnitude of the acceleration of the train is twice the magnitude of the deceleration.

1. On the axes below, sketch a speed-time graph to illustrate the motion of the train as it moves from station \(A\) to station \(B\). \includegraphics[max width=\textwidth, alt={}, center]{84c0eead-0a87-4d87-b33d-794a94bb466c-02_670_1422_813_312} If you need to redraw your graph, use the axes on page 3 Stations \(A\) and \(B\) are 4.8 km apart.
2. Find the value of \(T\)
3. Find the acceleration of the train during the first \(T\) seconds of its motion. Only use these axes if you need to redraw your graph. \({ } _ { O } ^ { \substack { \text { speed } \\ \left( \mathrm { ms } ^ { - 1 } \right) } }\)

5. \begin{figure}[h] \end{figure} Three points \(P , Q\) and \(R\) are on a horizontal road where \(P Q R\) is a straight line.
The point \(Q\) is between \(P\) and \(R\), with \(P Q = 6 x\) metres and \(Q R = 5 x\) metres, as shown in Figure 2. A vehicle moves along the road from \(P\) to \(Q\) with constant acceleration.
The vehicle is modelled as a particle.
At time \(t = 0\), the vehicle passes \(P\) with speed \(u \mathrm {~ms} ^ { - 1 }\) At time \(t = 12 \mathrm {~s}\), the vehicle passes \(Q\) with speed \(2 u \mathrm {~ms} ^ { - 1 }\) Using the model,

1. show that \(x = 3 u\) As the vehicle passes \(Q\), the acceleration of the vehicle changes instantaneously to \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The vehicle continues to move with a constant acceleration of \(1.5 \mathrm {~ms} ^ { - 2 }\) and passes \(R\) with speed \(3 u \mathrm {~ms} ^ { - 1 }\) Using the model,
2. find the value of \(u\),
3. find the distance travelled by the vehicle during the first 14 seconds after passing \(P\)

1. A particle is projected vertically upwards from a point \(A\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

The point \(A\) is 2.5 m vertically above the point \(B\).
Point \(B\) lies on horizontal ground.
The particle moves freely under gravity until it hits the ground at \(B\) with speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) After hitting the ground the particle does not rebound.

1. Find the value of \(V\).
2. Find the time taken for the particle to reach \(B\). The point \(C\) is 10 m vertically above \(A\).
3. Find the length of time for which the particle is above \(C\).
4. Sketch a speed-time graph for the motion of the particle from projection to the instant that it reaches \(B\). (No further calculations are required.)

3. A cyclist starts from rest at the point \(O\) on a straight horizontal road. The cyclist moves along the road with constant acceleration \(2 \mathrm {~ms} ^ { - 2 }\) for 4 seconds and then continues to move along the road at constant speed. At the instant when the cyclist stops accelerating, a motorcyclist starts from rest at the point \(O\) and moves along the road with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in the same direction as the cyclist. The motorcyclist has been moving for \(T\) seconds when she overtakes the cyclist.

1. Sketch, on the same axes, a speed-time graph for the motion of the cyclist and a speed-time graph for the motion of the motorcyclist, to the time when the motorcyclist overtakes the cyclist.
2. Find, giving your answer to 1 decimal place, the value of \(T\).

8. Two trams, tram \(A\) and tram \(B\), run on parallel straight horizontal tracks. Initially the two trams are at rest in the depot and level with each other. At time \(t = 0 , \operatorname { tram } A\) starts to move. Tram \(A\) moves with constant acceleration \(2 \mathrm {~ms} ^ { - 2 }\) for 5 seconds and then continues to move along the track at constant speed. At time \(t = 20\) seconds, tram \(B\) starts from rest and moves in the same direction as tram \(A\). Tram \(B\) moves with constant acceleration \(3 \mathrm {~ms} ^ { - 2 }\) for 4 seconds and then continues to move along the track at constant speed. The trams are modelled as particles.

1. Sketch, on the same axes, a speed-time graph for the motion of tram \(A\) and a speed-time graph for the motion of tram \(B\), from \(t = 0\) to the instant when tram \(B\) overtakes \(\operatorname { tram } A\). At the instant when the two trams are moving with the same speed, \(\operatorname { tram } A\) is \(d\) metres in front of tram \(B\).
2. Find the value of \(d\).
3. Find the distance of the trams from the depot at the instant when tram \(B\) overtakes \(\operatorname { tram } A\). \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-32_2647_1835_118_116}
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7. Two small children, Ajaz and Beth, are running a 100 m race along a straight horizontal track. They both start from rest, leaving the start line at the same time. Ajaz accelerates at \(0.8 \mathrm {~ms} ^ { - 2 }\) up to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then maintains this speed until he crosses the finish line. Beth accelerates at \(1 \mathrm {~ms} ^ { - 2 }\) for \(T\) seconds and then maintains a constant speed until she crosses the finish line. Ajaz and Beth cross the finish line at the same time.

1. Sketch, on the same axes, a speed-time graph for each child, from the instant when they leave the start line to the instant when they cross the finish line.
2. Find the time taken by Ajaz to complete the race.
3. Find the value of \(T\)
4. Find the difference in the speeds of the two children as they cross the finish line.