Two vehicles: overtaking or meeting (algebraic)

7 A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 36 m . The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(A B\) is 210 m .

1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). 24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
2. Find the time that it takes from when the cyclist starts until the car overtakes her.

4 A runner sets off from a point \(P\) at time \(t = 0\), where \(t\) is in seconds. The runner starts from rest and accelerates at \(1.2 \mathrm {~ms} ^ { - 2 }\) for 5 s . For the next 12 s the runner moves at constant speed before decelerating uniformly over a period of 3 s , coming to rest at \(Q\). A cyclist sets off from \(P\) at time \(t = 10\) and accelerates uniformly for 10 s , before immediately decelerating uniformly to rest at \(Q\) at time \(t = 30\).

1. Sketch the velocity-time graph for the runner and show that the distance \(P Q\) is 96 m . \includegraphics[max width=\textwidth, alt={}, center]{007ccd92-79ba-409a-97e8-a4cf1f0a6cc5-06_821_1451_708_388}
2. Find the magnitude of the acceleration of the cyclist.

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 \(T\).

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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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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6. Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. Car \(A\) is moving with uniform acceleration \(0.4 \mathrm {~ms} ^ { - 2 }\) and car \(B\) is moving with uniform acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At the instant when \(B\) is 200 m behind \(A\), the speed of \(A\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(44 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(B\) when it overtakes \(A\).
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6. \begin{figure}[h] \end{figure} Two cars, \(A\) and \(B\), move on parallel straight horizontal tracks. Initially \(A\) and \(B\) are both at rest with \(A\) at the point \(P\) and \(B\) at the point \(Q\), as shown in Figure 2. At time \(t = 0\) seconds, \(A\) starts to move with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 3.5 s , reaching a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Car \(A\) then moves with constant speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(a\). Car \(B\) also starts to move at time \(t = 0\) seconds, in the same direction as car \(A\). Car \(B\) moves with a constant acceleration of \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = T\) seconds, \(B\) overtakes \(A\). At this instant \(A\) is moving with constant speed.
2. On a diagram, sketch, on the same axes, a speed-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and a speed-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\).
3. Find the value of \(T\).
4. Find the distance of car \(B\) from the point \(Q\) when \(B\) overtakes \(A\).
5. On a new diagram, sketch, on the same axes, an acceleration-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and an acceleration-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
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11 A car starts from rest at a set of traffic lights and moves along a straight road with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). A motorcycle, travelling parallel to the car with constant speed \(16 \mathrm {~ms} ^ { - 1 }\), passes the same traffic lights exactly 1.5 seconds after the car starts to move. The time after the car starts to move is denoted by \(t\) seconds.

1. Determine the two values of \(t\) at which the car and motorcycle are the same distance from the traffic lights. These two values of \(t\) are denoted by \(t _ { 1 }\) and \(t _ { 2 }\), where \(t _ { 1 } < t _ { 2 }\).
2. Describe the relative positions of the car and the motorcycle when \(t _ { 1 } < t < t _ { 2 }\).
3. Determine the maximum distance between the car and the motorcycle when \(t _ { 1 } < t < t _ { 2 }\). \section*{END OF QUESTION PAPER}

9 Two trains are travelling in the same direction on parallel straight tracks and train A overtakes train B . At time \(t\) seconds after the front of train A overtakes the front of train B the velocities of trains A and B are \(v _ { \mathrm { A } } \mathrm { m } \mathrm { s } ^ { - 1 }\) and \(v _ { \mathrm { B } } \mathrm { ms } ^ { - 1 }\) respectively. The velocity of train A is modelled by \(\mathrm { v } _ { \mathrm { A } } = 25 - 0.6 \mathrm { t }\). The velocity-time graph of train A is shown below. \includegraphics[max width=\textwidth, alt={}, center]{b5c47a93-ce43-4aa1-ba7f-fbb650523373-5_664_1399_550_242}

1. A student argues that the speed of train A changes by \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 30 seconds so its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Comment on the validity of the student's argument.
2. When the front of train A overtakes the front of train B , train B has a velocity of \(10 \mathrm {~ms} ^ { - 1 }\). The acceleration of train \(B\) is constant and is modelled as \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Write down the equation for \(v _ { \mathrm { B } }\) in terms of \(t\) that models the velocity of train B .
3. Draw the velocity-time graph of train B on the copy of the diagram in the Printed Answer Booklet.
4. Determine the distance between the fronts of the trains at the time when the trains are travelling at the same velocity.
5. Explain why the model for train A would not be valid for large values of \(t\).

8 A bus is travelling along a straight road at \(5.4 \mathrm {~ms} ^ { - 1 }\). At \(t = 0\), as the bus passes a boy standing on the pavement, the boy starts running in the same direction as the bus, accelerating at \(1.2 \mathrm {~ms} ^ { - 2 }\) from rest for 5 s . He then runs at constant speed until he catches up with the bus.

1. The diagram in the Printed Answer Booklet shows the velocity-time graph for the bus. Draw the velocity-time graph for the boy on this diagram.
2. Determine the time at which the boy is running at the same speed as the bus.
3. Find the maximum distance between the bus and the boy.
4. Find the distance the boy has run when he catches up with the bus.

5. A car and a motorbike are at rest adjacent to one another at a set of traffic lights on a long, straight stretch of road. They set off simultaneously at time \(t = 0\). The motorcyclist accelerates uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) until he reaches a speed of \(30 \mathrm {~ms} ^ { - 1 }\) which he then maintains. The car driver accelerates uniformly for 9 seconds until she reaches \(36 \mathrm {~ms} ^ { - 1 }\) and then remains at this speed.

1. Find the acceleration of the car.
2. Draw on the same diagram speed-time graphs to illustrate the movements of both vehicles.
3. Find the value of \(t\) when the car again draws level with the motorcyclist.