Multi-phase journey: find unknown speed or time

4 A car starts from rest and moves in a straight line with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 50 m . The car then travels with constant velocity for 500 m for a period of 25 s , before decelerating to rest. The magnitude of this deceleration is \(2 a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Sketch the velocity-time graph for the motion of the car. \includegraphics[max width=\textwidth, alt={}, center]{55090630-1413-45cd-8201-4d58662db6bd-05_533_1155_534_534}
2. Find the value of \(a\).
3. Find the total time for which the car is in motion.

6 An elevator is pulled vertically upwards by a cable. The elevator accelerates at \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 5 s , then travels at constant speed for 25 s . The elevator then decelerates at \(0.2 \mathrm {~ms} ^ { - 2 }\) until it comes to rest.

1. Find the greatest speed of the elevator and hence draw a velocity-time graph for the motion of the elevator.
2. Find the total distance travelled by the elevator.
   The mass of the elevator is 1200 kg and there is a crate of mass \(m \mathrm {~kg}\) resting on the floor of the elevator.
3. Given that the tension in the cable when the elevator is decelerating is 12250 N , find the value of \(m\).
4. Find the greatest magnitude of the force exerted on the crate by the floor of the elevator, and state its direction.

1 A bus moves from rest with constant acceleration for 12 s . It then moves with constant speed for 30 s before decelerating uniformly to rest in a further 6 s . The total distance travelled is 585 m .

1. Find the constant speed of the bus.
2. Find the magnitude of the deceleration.

5 A train starts from rest at a station \(A\) and travels in a straight line to station \(B\), where it comes to rest. The train moves with constant acceleration \(0.025 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for the first 600 s , with constant speed for the next 2600 s , and finally with constant deceleration \(0.0375 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the total time taken for the train to travel from \(A\) to \(B\).
2. Sketch the velocity-time graph for the journey and find the distance \(A B\).
3. The speed of the train \(t\) seconds after leaving \(A\) is \(7.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). State the possible values of \(t\).

3 A train travels between two stations, \(A\) and \(B\). The train starts from rest at \(A\) and accelerates at a constant rate for \(T\) s until it reaches a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then travels at this constant speed before decelerating at a constant rate, coming to rest at \(B\). The magnitude of the train's deceleration is twice the magnitude of its acceleration. The total time taken for the journey is 180 s .

1. Sketch the velocity-time graph for the train's journey from \(A\) to \(B\). \includegraphics[max width=\textwidth, alt={}, center]{3d7f53af-dbf2-499b-9966-ae85514cef02-04_496_857_516_685}
2. Find an expression, in terms of \(T\), for the length of time for which the train is travelling with constant speed.
3. The distance from \(A\) to \(B\) is 3300 m . Find how far the train travels while it is decelerating.

6 A train travels from \(A\) to \(B\), a distance of 20000 m , taking 1000 s . The journey has three stages. In the first stage the train starts from rest at \(A\) and accelerates uniformly until its speed is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). In the second stage the train travels at constant speed \(V _ { \mathrm { m } } { } ^ { - 1 }\) for 600 s . During the third stage of the journey the train decelerates uniformly, coming to rest at \(B\).

1. Sketch the velocity-time graph for the train's journey.
2. Find the value of \(V\).
3. Given that the acceleration of the train during the first stage of the journey is \(0.15 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), find the distance travelled by the train during the third stage of the journey. \(7 \quad\) A particle \(P\) is held at rest at a fixed point \(O\) and then released. \(P\) falls freely under gravity until it reaches the point \(A\) which is 1.25 m below \(O\).
4. Find the speed of \(P\) at \(A\) and the time taken for \(P\) to reach \(A\). The particle continues to fall, but now its downward acceleration \(t\) seconds after passing through \(A\) is \(( 10 - 0.3 t ) \mathrm { m } \mathrm { s } ^ { - 2 }\).
5. Find the total distance \(P\) has fallen, 3 s after being released from \(O\).

5 A car travels in a straight line from \(A\) to \(B\), a distance of 12 km , taking 552 seconds. The car starts from rest at \(A\) and accelerates for \(T _ { 1 } \mathrm {~s}\) at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reaching a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then continues to move at \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T _ { 2 } \mathrm {~s}\). It then decelerates for \(T _ { 3 } \mathrm {~s}\) at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).

1. Sketch the velocity-time graph for the motion and express \(T _ { 1 }\) and \(T _ { 3 }\) in terms of \(V\).
2. Express the total distance travelled in terms of \(V\) and show that \(13 V ^ { 2 } - 3312 V + 72000 = 0\). Hence find the value of \(V\).

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 \(V\).

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 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 \(C\).

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. A car travelling along a straight horizontal road takes 170 s to travel between two sets of traffic lights at \(A\) and \(B\) which are 2125 m apart. The car starts from rest at \(A\) and moves with constant acceleration until it reaches a speed of \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then maintains this speed before moving with constant deceleration, coming to rest at \(B\). The magnitude of the deceleration is twice the magnitude of the acceleration.

1. Sketch, in the space below, a speed-time graph for the motion of the car between \(A\) and \(B\).
2. Find the deceleration of the car.

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.

5. \begin{figure}[h] \end{figure} The speed-time graph in Figure 2 illustrates the motion of a car travelling along a straight horizontal road.
At time \(t = 0\), the car starts from rest and accelerates uniformly for 30 s until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The car then travels at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until time \(t = T\) seconds.

1. Show that the distance travelled by the car between \(t = 0\) and \(t = T\) seconds is \(V ( T - 15 )\) metres. A motorbike also travels along the same road.
   At time \(t = T\) seconds, the distance travelled by each vehicle is the same.
2. Find the value of \(T\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-15_643_1266_1882_402} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}

7. A train moves on a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration \(1 \mathrm {~ms} ^ { - 2 }\) until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train maintains this speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next \(T\) seconds before slowing down with constant deceleration \(0.5 \mathrm {~ms} ^ { - 2 }\), coming to rest at \(B\). The journey from \(A\) to \(B\) takes 180 s and the distance between the stations is 4800 m .

1. Sketch a speed-time graph for the motion of the train from \(A\) to \(B\).
2. Show that \(T = 180 - 3 V\).
3. Find the value of \(V\).

3. A car moves along a horizontal straight road, passing two points \(A\) and \(B\). At \(A\) the speed of the car is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the driver passes \(A\), he sees a warning sign \(W\) ahead of him, 120 m away. He immediately applies the brakes and the car decelerates with uniform deceleration, reaching \(W\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At \(W\), the driver sees that the road is clear. He then immediately accelerates the car with uniform acceleration for 16 s to reach a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 } ( V > 15 )\). He then maintains the car at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Moving at this constant speed, the car passes \(B\) after a further 22 s .

1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car as it moves from \(A\) to \(B\).
2. Find the time taken for the car to move from \(A\) to \(B\). The distance from \(A\) to \(B\) is 1 km .
3. Find the value of \(V\).

5 \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-3_394_789_251_639} The diagram shows the ( \(t , v\) ) graph of an athlete running in a straight line on a horizontal track in a 100 m race. He starts from rest and has constant acceleration until he reaches a speed of \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = T\). He maintains this constant speed until he decelerates at a constant rate of \(1.75 \mathrm {~ms} ^ { - 2 }\) for the final 4 s of the race. He completes the race in 10 s .

1. Calculate \(T\). The athlete races against a robot which has a displacement from the starting line of \(\left( 3 t ^ { 2 } - 0.2 t ^ { 3 } \right) \mathrm { m }\), at time \(t \mathrm {~s}\) after the start of the race.
2. Show that the speed of the robot is \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = 5\).
3. Find the value of \(t\) for which the decelerations of the robot and the athlete are equal.
4. Verify that the athlete and the robot reach the finish line simultaneously.

1. A train travels along a straight horizontal track from station \(P\) to station \(Q\).

In a model of the motion of the train, at time \(t = 0\) the train starts from rest at \(P\), and moves with constant acceleration until it reaches its maximum speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The train then travels at this constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before finally moving with constant deceleration until it comes to rest at \(Q\). The time spent decelerating is four times the time spent accelerating.
The journey from \(P\) to \(Q\) takes 700 s .
Using the model,

1. sketch a speed-time graph for the motion of the train between the two stations \(P\) and \(Q\). The distance between the two stations is 15 km .
   Using the model,
2. show that the time spent accelerating by the train is 40 s ,
3. find the acceleration, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of the train,
4. find the speed of the train 572s after leaving \(P\).
5. State one limitation of the model which could affect your answers to parts (b) and (c).

1. \begin{figure}[h] \end{figure} Two children, Pat \(( P )\) and Sam \(( S )\), run a race along a straight horizontal track.
Both children start from rest at the same time and cross the finish line at the same time.
In a model of the motion:
Pat accelerates at a constant rate from rest for 5 s until reaching a speed of \(4 \mathrm {~ms} ^ { - 1 }\) and then maintains a constant speed of \(4 \mathrm {~ms} ^ { - 1 }\) until crossing the finish line. Sam accelerates at a constant rate of \(1 \mathrm {~ms} ^ { - 2 }\) from rest until reaching a speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then maintains a constant speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until crossing the finish line. Both children take 27.5 s to complete the race.
The velocity-time graphs shown in Figure 1 describe the model of the motion of each child from the instant they start to the instant they cross the finish line together. Using the model,

1. explain why the areas under the two graphs are equal,
2. find the acceleration of Pat during the first 5 seconds,
3. find, in metres, the length of the race,
4. find the value of \(X\), giving your answer to 3 significant figures.

10 A cyclist starts from rest and moves with constant acceleration along a straight horizontal road. The cyclist reaches a speed of \(6 \mathrm {~ms} ^ { - 1 }\) in 25 seconds. The cyclist then moves with constant acceleration \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until the speed is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist then moves with constant deceleration until coming to rest. The total time for the cyclist's journey is 150 seconds.

1. Sketch a velocity-time graph to represent the cyclist's motion.
2. Find the acceleration during the first 25 seconds of the cyclist's motion. The cyclist takes \(T\) seconds to decelerate from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until coming to rest.
3. Determine the value of \(T\).
4. Determine the average speed for the cyclist's journey.

9 A cyclist travels along a straight horizontal road between house \(A\) and house \(B\). The cyclist starts from rest at \(A\) and moves with constant acceleration for 20 seconds, reaching a velocity of \(15 \mathrm {~ms} ^ { - 1 }\). The cyclist then moves at this constant velocity before decelerating at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).

1. Find the time, in seconds, for which the cyclist is decelerating.
2. Sketch a velocity-time graph for the motion of the cyclist between \(A\) and \(B\). [Your sketch need not be drawn to scale; numerical values need not be shown.] The total distance between \(A\) and \(B\) is 1950 m .
3. Find the time, in seconds, for which the cyclist is moving at constant velocity.