Multi-stage motion with velocity-time graph given

A question is this type if and only if it provides a velocity-time graph showing multiple stages of motion and requires finding unknown parameters, distances, or times from the graph.

29 questions · Moderate -0.5

3.02c Interpret kinematic graphs: gradient and area
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CAIE M1 2021 March Q4
6 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{a96ca3b4-6d35-4512-a0a1-3f28443fd051-06_661_1529_260_306} An elevator moves vertically, supported by a cable. The diagram shows a velocity-time graph which models the motion of the elevator. The graph consists of 7 straight line segments. The elevator accelerates upwards from rest to a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 1.5 s and then travels at this speed for 4.5 s , before decelerating to rest over a period of 1 s . The elevator then remains at rest for 6 s , before accelerating to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards over a period of 2 s . The elevator travels at this speed for a period of 5 s , before decelerating to rest over a period of 1.5 s .
  1. Find the acceleration of the elevator during the first 1.5 s .
  2. Given that the elevator starts and finishes its journey on the ground floor, find \(V\).
  3. The combined weight of the elevator and passengers on its upward journey is 1500 kg . Assuming that there is no resistance to motion, find the tension in the elevator cable on its upward journey when the elevator is decelerating.
CAIE M1 2020 November Q4
5 marks Moderate -0.5
4
[diagram]
The diagram shows a velocity-time graph which models the motion of a car. The graph consists of four straight line segments. The car accelerates at a constant rate of \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of \(T \mathrm {~s}\). It then decelerates at a constant rate for 5 seconds before travelling at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 27.5 s . The car then decelerates to rest at a constant rate over a period of 5 s .
  1. Find \(T\).
  2. Given that the distance travelled up to the point at which the car begins to move with constant speed is one third of the total distance travelled, find \(V\).
CAIE M1 2021 November Q1
4 marks Moderate -0.3
1 \includegraphics[max width=\textwidth, alt={}, center]{083d3e44-1e42-461f-aa8d-a1a22047a47e-02_611_1351_260_397} The diagram shows a velocity-time graph which models the motion of a car. The graph consists of six straight line segments. The car accelerates from rest to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 5 s , and then travels at this speed for a further 20 s . The car then decelerates to a speed of \(6 \mathrm {~ms} ^ { - 1 }\) over a period of 5 s . This speed is maintained for a further \(( T - 30 ) \mathrm { s }\). The car then accelerates again to a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of \(( 50 - T ) \mathrm { s }\), before decelerating to rest over a period of 10 s .
  1. Given that during the two stages of the motion when the car is accelerating, the accelerations are equal, find the value of \(T\).
  2. Find the total distance travelled by the car during the motion.
CAIE M1 2023 November Q3
8 marks Moderate -0.8
3 \includegraphics[max width=\textwidth, alt={}, center]{f1f33ef0-0d4d-4a4a-aadb-28de8dc0ea8d-04_666_1278_280_424} The diagram shows the velocity-time graph for the motion of a bus. The bus starts from rest and accelerates uniformly for 8 seconds until it reaches a speed of \(12.6 \mathrm {~ms} ^ { - 1 }\). The bus maintains this speed for 40 seconds. It then decelerates uniformly in two stages. Between 48 and 62 seconds the bus decelerates at \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and between 62 and 70 seconds it decelerates at \(2 a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until coming to rest.
  1. Find the distance covered by the bus in the first 8 seconds.
  2. Find the value of \(a\).
  3. Find the average speed of the bus for the whole journey.
CAIE M1 2016 June Q4
7 marks Moderate -0.8
4 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-2_522_959_1692_593} A sprinter runs a race of 400 m . His total time for running the race is 52 s . The diagram shows the velocity-time graph for the motion of the sprinter. He starts from rest and accelerates uniformly to a speed of \(8.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 s . The sprinter maintains a speed of \(8.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 36 s , and he then decelerates uniformly to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of the race.
  1. Calculate the distance covered by the sprinter in the first 42 s of the race.
  2. Show that \(V = 7.84\).
  3. Calculate the deceleration of the sprinter in the last 10 s of the race.
CAIE M1 2010 November Q6
9 marks Standard +0.3
6 \includegraphics[max width=\textwidth, alt={}, center]{881993e1-71ea-4801-bfc8-40c17a1387a9-3_579_1518_258_315} The diagram shows the velocity-time graph for a particle \(P\) which travels on a straight line \(A B\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of \(P\) at time \(t \mathrm {~s}\). The graph consists of five straight line segments. The particle starts from rest when \(t = 0\) at a point \(X\) on the line between \(A\) and \(B\) and moves towards \(A\). The particle comes to rest at \(A\) when \(t = 2.5\).
  1. Given that the distance \(X A\) is 4 m , find the greatest speed reached by \(P\) during this stage of the motion. In the second stage, \(P\) starts from rest at \(A\) when \(t = 2.5\) and moves towards \(B\). The distance \(A B\) is 48 m . The particle takes 12 s to travel from \(A\) to \(B\) and comes to rest at \(B\). For the first 2 s of this stage \(P\) accelerates at \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reaching a velocity of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find
  2. the value of \(V\),
  3. the value of \(t\) at which \(P\) starts to decelerate during this stage,
  4. the deceleration of \(P\) immediately before it reaches \(B\). \(7 \quad\) A particle \(P\) travels in a straight line. It passes through the point \(O\) of the line with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t = 0\), where \(t\) is in seconds. \(P\) 's velocity after leaving \(O\) is given by $$\left( 0.002 t ^ { 3 } - 0.12 t ^ { 2 } + 1.8 t + 5 \right) \mathrm { m } \mathrm {~s} ^ { - 1 }$$ The velocity of \(P\) is increasing when \(0 < t < T _ { 1 }\) and when \(t > T _ { 2 }\), and the velocity of \(P\) is decreasing when \(T _ { 1 } < t < T _ { 2 }\).
  5. Find the values of \(T _ { 1 }\) and \(T _ { 2 }\) and the distance \(O P\) when \(t = T _ { 2 }\).
  6. Find the velocity of \(P\) when \(t = T _ { 2 }\) and sketch the velocity-time graph for the motion of \(P\).
CAIE M1 2016 November Q5
8 marks Standard +0.3
5 \includegraphics[max width=\textwidth, alt={}, center]{a92f97e2-343f-4cac-ae38-f18a4ad49055-3_574_1205_260_470} The diagram shows a velocity-time graph which models the motion of a cyclist. The graph consists of five straight line segments. The cyclist accelerates from rest to a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 10 s , and then travels at this speed for a further 20 s . The cyclist then descends a hill, accelerating to speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over a period of 10 s . This speed is maintained for a further 30 s . The cyclist then decelerates to rest over a period of 20 s .
  1. Find the acceleration of the cyclist during the first 10 seconds.
  2. Show that the total distance travelled by the cyclist in the 90 seconds of motion may be expressed as \(( 45 V + 150 ) \mathrm { m }\). Hence find \(V\), given that the total distance travelled by the cyclist is 465 m .
  3. The combined mass of the cyclist and the bicycle is 80 kg . The cyclist experiences a constant resistance to motion of 20 N . Use an energy method to find the vertical distance which the cyclist descends during the downhill section from \(t = 30\) to \(t = 40\), assuming that the cyclist does no work during this time.
CAIE M1 2017 November Q4
7 marks Moderate -0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{db1b5f31-1a41-44dd-ae9a-0c67336997eb-05_600_1155_262_497} The diagram shows the velocity-time graph of a particle which moves in a straight line. The graph consists of 5 straight line segments. The particle starts from rest at a point \(A\) at time \(t = 0\), and initially travels towards point \(B\) on the line.
  1. Show that the acceleration of the particle between \(t = 3.5\) and \(t = 6\) is \(- 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
  2. The acceleration of the particle between \(t = 6\) and \(t = 10\) is \(7.5 \mathrm {~ms} ^ { - 2 }\). When \(t = 10\) the velocity of the particle is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(V\).
  3. The particle comes to rest at \(B\) at time \(T\) s. Given that the total distance travelled by the particle between \(t = 0\) and \(t = T\) is 100 m , find the value of \(T\).
OCR M1 2010 June Q5
9 marks Moderate -0.8
5 \includegraphics[max width=\textwidth, alt={}, center]{4b703cf9-b3d3-4210-b57b-89136595f8a5-03_508_1397_255_374} The diagram shows the ( \(t , v\) ) graph for a lorry delivering waste to a recycling centre. The graph consists of six straight line segments. The lorry reverses in a straight line from a stationary position on a weighbridge before coming to rest. It deposits its waste and then moves forwards in a straight line accelerating to a maximum speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It maintains this speed for 4 s and then decelerates, coming to rest at the weighbridge.
  1. Calculate the distance from the weighbridge to the point where the lorry deposits the waste.
  2. Calculate the time which elapses between the lorry leaving the weighbridge and returning to it.
  3. Given that the acceleration of the lorry when it is moving forwards is \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), calculate its final deceleration.
OCR M1 2013 June Q4
10 marks Standard +0.3
4 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-3_298_540_262_735} The diagram shows the \(( t , v )\) graph of a car moving along a straight road, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the car at time \(t \mathrm {~s}\) after it passes through the point \(A\). The car passes through \(A\) with velocity \(18 \mathrm {~ms} ^ { - 1 }\), and moves with constant acceleration \(2.4 \mathrm {~ms} ^ { - 2 }\) until \(t = 5\). The car subsequently moves with constant velocity until it is 300 m from \(A\). When the car is more than 300 m from \(A\), it has constant deceleration \(6 \mathrm {~ms} ^ { - 2 }\), until it comes to rest.
  1. Find the greatest speed of the car.
  2. Calculate the value of \(t\) for the instant when the car begins to decelerate.
  3. Calculate the distance from \(A\) of the car when it is at rest.
Edexcel M1 2024 October Q4
Moderate -0.8
  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}
CAIE M1 2023 June Q4
8 marks Moderate -0.3
\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]
CAIE M1 2005 June Q6
9 marks Moderate -0.8
\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]
CAIE M1 2010 June Q2
5 marks Easy -1.2
\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]
CAIE M1 2010 June Q2
5 marks Easy -1.2
\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 \(8 \text{ s}\) while cutting and then takes \(3 \text{ s}\) to return to its starting position. Find
  1. the acceleration of the tool during the first \(2 \text{ s}\) 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]
CAIE M1 2018 June Q1
5 marks Easy -1.2
\includegraphics{figure_1} The diagram shows the velocity-time graph for a train which travels from rest at one station to rest at the next station. The graph consists of three straight line segments. The distance between the two stations is 9040 m.
  1. Find the acceleration of the train during the first 40 s. [1]
  2. Find the length of time for which the train is travelling at constant speed. [2]
  3. Find the distance travelled by the train while it is decelerating. [2]
CAIE M1 2019 March Q5
7 marks Moderate -0.8
\includegraphics{figure_5} The velocity of a particle moving in a straight line is \(v\) m s\(^{-1}\) at time \(t\) seconds after leaving a fixed point \(O\). The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = 16\). The graph consists of five straight line segments. The acceleration of the particle from \(t = 0\) to \(t = 3\) is \(\frac{7}{3}\) m s\(^{-2}\). The velocity of the particle at \(t = 5\) is \(7\) m s\(^{-1}\) and it comes to instantaneous rest at \(t = 8\). The particle then comes to rest again at \(t = 16\). The minimum velocity of the particle is \(V\) m s\(^{-1}\).
  1. Find the distance travelled by the particle in the first \(8\) s of its motion. [3]
  2. Given that when the particle comes to rest at \(t = 16\) its displacement from \(O\) is \(32\) m, find the value of \(V\). [4]
CAIE M1 2018 November Q3
7 marks Moderate -0.8
\includegraphics{figure_3} The velocity of a particle moving in a straight line is \(v\) m s\(^{-1}\) at time \(t\) seconds. The diagram shows a velocity-time graph which models the motion of the particle from \(t = 0\) to \(t = T\). The graph consists of four straight line segments. The particle reaches its maximum velocity \(V\) m s\(^{-1}\) at \(t = 10\).
  1. Find the acceleration of the particle during the first \(2\) seconds. [1]
  2. Find the value of \(V\). [2]
At \(t = 6\), the particle is instantaneously at rest at the point \(A\). At \(t = T\), the particle comes to rest at the point \(B\). At \(t = 0\) the particle starts from rest at a point one third of the way from \(A\) to \(B\).
  1. Find the distance \(AB\) and hence find the value of \(T\). [4]
CAIE M1 2019 November Q2
5 marks Moderate -0.8
\includegraphics{figure_2} The diagram shows a velocity-time graph which models the motion of a tractor. The graph consists of four straight line segments. The tractor passes a point \(O\) at time \(t = 0\) with speed \(U\) m s\(^{-1}\). The tractor accelerates to a speed of \(V\) m s\(^{-1}\) over a period of 5 s, and then travels at this speed for a further 25 s. The tractor then accelerates to a speed of 12 m s\(^{-1}\) over a period of 5 s. The tractor then decelerates to rest over a period of 15 s.
  1. Given that the acceleration of the tractor between \(t = 30\) and \(t = 35\) is 0.8 m s\(^{-2}\), find the value of \(V\). [2]
  2. Given also that the total distance covered by the tractor in the 50 seconds of motion is 375 m, find the value of \(U\). [3]
Edexcel M1 2005 January Q3
9 marks Moderate -0.8
\includegraphics{figure_2} 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 m s\(^{-1}\) in 4 s. The speed of 9 m s\(^{-1}\) is maintained for 16 s and she then decelerates uniformly to a speed of \(u\) m 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]
  2. the value of \(u\), [4]
  3. the deceleration of the sprinter in the last 5 s of the race. [3]
Edexcel M1 2013 January Q5
15 marks Moderate -0.8
\includegraphics{figure_4} The velocity-time graph in Figure 4 represents the journey of a train \(P\) travelling along a straight horizontal track between two stations which are 1.5 km apart. The train \(P\) leaves the first station, accelerating uniformly from rest for 300 m until it reaches a speed of 30 m s\(^{-1}\). The train then maintains this speed for 7 seconds before decelerating uniformly at 1.25 m s\(^{-2}\), coming to rest at the next station.
  1. Find the acceleration of \(P\) during the first 300 m of its journey. [2]
  2. Find the value of \(T\). [5]
A second train \(Q\) completes the same journey in the same total time. The train leaves the first station, accelerating uniformly from rest until it reaches a speed of \(V\) m s\(^{-1}\) and then immediately decelerates uniformly until it comes to rest at the next station.
  1. Sketch on the diagram above, a velocity-time graph which represents the journey of train \(Q\). [2]
  2. Find the value of \(V\). [6]
Edexcel M1 Q6
12 marks Moderate -0.8
The diagram shows the speed-time graph for a particle during a period of \(9T\) seconds. \includegraphics{figure_6}
  1. If \(T = 5\), find
    1. the acceleration for each section of the motion, [2 marks]
    2. the total distance travelled by the particle. [2 marks]
  2. Sketch, for this motion,
    1. an acceleration-time graph, [2 marks]
    2. a displacement-time graph. [2 marks]
  3. Calculate the value of \(T\) for which the distance travelled over the \(9T\) seconds is 3.708 km. [4 marks]
Edexcel M1 Q6
14 marks Standard +0.3
The diagram shows the velocity-time graph for a cyclist's journey. Each section has constant acceleration or deceleration and the three sections are of equal duration \(x\) seconds each. \includegraphics{figure_6} Given that the total distance travelled is \(792\) m,
  1. find the value of \(x\) and the acceleration for the first section of the journey. [6 marks]
Another cyclist covers the same journey in three sections of equal duration, accelerating at \(\frac{1}{11} \text{ ms}^{-2}\) for the first section, travelling at constant speed for the second section and decelerating at \(\frac{1}{11} \text{ ms}^{-2}\) for the third section.
  1. Find the time taken by this cyclist to complete the journey. [6 marks]
  2. Show that the maximum speeds of both cyclists are the same. [2 marks]
Edexcel M1 Q6
13 marks Moderate -0.3
The velocity-time graph illustrates the motion of a particle which accelerates from rest to 8 ms\(^{-1}\) in \(x\) seconds and then to 24 ms\(^{-1}\) in a further 4 seconds. It then travels at a constant speed for another \(y\) seconds before decelerating to 12 ms\(^{-1}\) over the next \(y\) seconds and then to rest in the final 7 seconds of its motion. \includegraphics{figure_6} Given that the total distance travelled by the particle is 496 m,
  1. show that \(2x + 21y = 195\). [4 marks]
Given also that the average speed of the particle during its motion is 15.5 ms\(^{-1}\),
  1. show that \(x + 2y = 21\). [3 marks]
  2. Hence find the values of \(x\) and \(y\). [3 marks]
  3. Write down the acceleration for each section of the motion. [3 marks]
OCR M1 Q6
12 marks Standard +0.3
\includegraphics{figure_6} The diagram shows the \((t, v)\) graph for the motion of a hoist used to deliver materials to different levels at a building site. The hoist moves vertically. The graph consists of straight line segments. In the first stage the hoist travels upwards from ground level for \(25\) s, coming to rest \(8\) m above ground level.
  1. Find the greatest speed reached by the hoist during this stage. [2]
The second stage consists of a \(40\) s wait at the level reached during the first stage. In the third stage the hoist continues upwards until it comes to rest \(40\) m above ground level, arriving \(135\) s after leaving ground level. The hoist accelerates at \(0.02\) m s\(^{-2}\) for the first \(40\) s of the third stage, reaching a speed of \(V\) m s\(^{-1}\). Find
  1. the value of \(V\), [3]
  2. the length of time during the third stage for which the hoist is moving at constant speed, [4]
  3. the deceleration of the hoist in the final part of the third stage. [3]