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

A particle \(A\), moving along a straight horizontal track with constant speed \(8\text{ms}^{-1}\), passes a fixed point \(O\). Four seconds later, another particle \(B\) passes \(O\), moving along a parallel track in the same direction as \(A\). Particle \(B\) has speed \(20\text{ms}^{-1}\) when it passes \(O\) and has a constant deceleration of \(2\text{ms}^{-2}\). \(B\) comes to rest when it returns to \(O\).

1. Find expressions, in terms of \(t\), for the displacement from \(O\) of each particle \(t\) seconds after \(B\) passes \(O\). [3]
2. Find the values of \(t\) when the particles are the same distance from \(O\). [3]
3. On the given axes, sketch the displacement-time graphs for both particles, for values of \(t\) from \(0\) to \(20\). [3] $$s \text{ (m)}$$ $$200$$ $$100$$ $$0 \quad 0 \quad 10 \quad 20 \quad t \text{ (s)}$$

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]

A bus travels between two stops, \(A\) and \(B\). The bus starts from rest at \(A\) and accelerates at a constant rate of \(a \text{ ms}^{-2}\) until it reaches a speed of \(16 \text{ ms}^{-1}\). It then travels at this constant speed before decelerating at a constant rate of \(0.75 \text{ ms}^{-2}\), coming to rest at \(B\). The total time for the journey is \(240\) s.

1. Sketch the velocity-time graph for the bus's journey from \(A\) to \(B\). [1]
2. Find an expression, in terms of \(a\), for the length of time that the bus is travelling with constant speed. [2]
3. Given that the distance from \(A\) to \(B\) is \(3000\) m, find the value of \(a\). [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]

\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]

\includegraphics{figure_6} A particle is projected vertically upward from ground level with speed \(u\) m s\(^{-1}\). The particle moves under gravity alone.

1. Find an expression for the maximum height reached by the particle. [3]

\includegraphics{figure_6b} The diagram shows a velocity-time graph for the motion of the particle.

1. Use the graph to find the value of \(u\). [2]
2. Find the time taken for the particle to return to ground level. [3]

A sprinter runs a race of \(200 \text{ m}\). His total time for running the race is \(20 \text{ s}\). He starts from rest and accelerates uniformly for \(6 \text{ s}\), reaching a speed of \(12 \text{ m s}^{-1}\). He maintains this speed for the next \(10 \text{ s}\), before decelerating uniformly to cross the finishing line with speed \(V \text{ m s}^{-1}\).

1. Find the distance travelled by the sprinter in the first \(16 \text{ s}\) of the race. Hence sketch a displacement-time graph for the \(20 \text{ s}\) of the sprinter's race. [6]
2. Find the value of \(V\). [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]

A particle \(P\) moves in a straight line starting from a point \(O\). The velocity \(v\text{ m s}^{-1}\) of \(P\) at time \(t\text{ s}\) is given by $$v = 12t - 4t^2 \quad \text{for } 0 \leqslant t \leqslant 2,$$ $$v = 16 - 4t \quad \text{for } 2 \leqslant t \leqslant 4.$$

1. Find the maximum velocity of \(P\) during the first \(2\text{ s}\). [3]
2. Determine, with justification, whether there is any instantaneous change in the acceleration of \(P\) when \(t = 2\). [2]
3. Sketch the velocity-time graph for \(0 \leqslant t \leqslant 4\). [3]
4. Find the distance travelled by \(P\) in the interval \(0 \leqslant t \leqslant 4\). [5]

Particles \(P\) and \(Q\) leave a fixed point \(A\) at the same time and travel in the same straight line. The velocity of \(P\) after \(t\) seconds is \(6(t - 3)\) m s\(^{-1}\) and the velocity of \(Q\) after \(t\) seconds is \((10 - 2t)\) m s\(^{-1}\).

1. Sketch, on the same axes, velocity-time graphs for \(P\) and \(Q\) for \(0 \leq t \leq 5\). [3]
2. Verify that \(P\) and \(Q\) meet after 5 seconds. [4]
3. Find the greatest distance between \(P\) and \(Q\) for \(0 \leq t \leq 5\). [4]

A particle \(P\) moves in a straight line starting from a point \(O\) and comes to rest \(35\) s later. At time \(t\) s after leaving \(O\), the velocity \(v\) m s\(^{-1}\) of \(P\) is given by $$v = \frac{4}{5}t^2 \quad 0 \leq t \leq 5,$$ $$v = 2t + 10 \quad 5 \leq t \leq 15,$$ $$v = a + bt^2 \quad 15 \leq t \leq 35,$$ where \(a\) and \(b\) are constants such that \(a > 0\) and \(b < 0\).

1. Show that the values of \(a\) and \(b\) are \(49\) and \(-0.04\) respectively. [3]
2. Sketch the velocity-time graph. [4]
3. Find the total distance travelled by \(P\) during the \(35\) s. [5]

\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]

\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]

A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of speed when the particle enters the water. The water exerts an upward force of 5.6 N on the particle when it is in the water.

1. Find the velocity of the particle at the instant when it reaches the surface of the water. [2]

1. Find the time that it takes from the instant when the particle enters the water until it comes to instantaneous rest in the water. You may assume that the tank is deep enough so that the particle does not reach the bottom of the tank. [4]

1. Sketch a velocity-time graph for the motion of the particle from the instant at which it is released until it comes to instantaneous rest in the water. [3]

\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]

A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration 0.5 m 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 \(AB\) is 210 m.

1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). [5]
2. Find the time that it takes from when the cyclist starts until the car overtakes her. [5]

24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration 4 m s\(^{-2}\) towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.

A particle moves in a straight line. Its displacement from a fixed point \(O\) at time \(t\) seconds is \(s\) metres, where \(s = t^3 - 9t^2 + 24t\).

1. Find expressions for the velocity \(v\) and acceleration \(a\) of the particle at time \(t\).
2. Find the values of \(t\) for which the particle is at rest.
3. Find the total distance travelled by the particle in the first \(6\) seconds.

[8]

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 until it reaches its maximum speed of 108 km h\(^{-1}\). The train then travels at this speed before it moves with constant deceleration coming to rest at \(B\). The journey from \(A\) to \(B\) takes 8 minutes.

1. Change 108 km h\(^{-1}\) into m s\(^{-1}\). [2]
2. Sketch a speed-time graph for the motion of the train between the two stations \(A\) and \(B\). [2]

Given that the distance between the two stations is 12 km and that the time spent decelerating is three times the time spent accelerating,

1. find the acceleration, in m s\(^{-2}\), of the train. [6]

A small stone is projected vertically upwards from the point \(O\) and moves freely under gravity. The point \(A\) is 3.6 m vertically above \(O\). When the stone first reaches \(A\), the stone is moving upwards with speed 11.2 m s\(^{-1}\). The stone is modelled as a particle.

1. Find the maximum height above \(O\) reached by the stone. [4]
2. Find the total time between the instant when the stone was projected from \(O\) and the instant when it returns to \(O\). [5]
3. Sketch a velocity-time graph to represent the motion of the stone from the instant when it passes through \(A\) moving upwards to the instant when it returns to \(O\). Show, on the axes, the coordinates of the points where your graph meets the axes. [4]

\includegraphics{figure_2} 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 \(a\) m s\(^{-2}\) for 3.5 s, reaching a speed of 14 m s\(^{-1}\). Car \(A\) then moves with constant speed 14 m s\(^{-1}\).

1. Find the value of \(a\). [2] Car \(B\) also starts to move at time \(t = 0\) seconds, in the same direction as car \(A\). Car \(B\) moves with constant acceleration of 3 m 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]
3. Find the value of \(T\). [8]
4. Find the distance of car \(B\) from the point \(Q\) when \(B\) overtakes \(A\). [1]
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 \(0 \leqslant t \leqslant T\). [3]

An athlete goes for a run along a straight horizontal road. Starting from rest, she accelerates at 0.6 m s\(^{-2}\) up to a speed of \(V\) m s\(^{-1}\). She then maintains this constant speed of \(V\) m s\(^{-1}\) before finally decelerating at 0.2 m s\(^{-2}\) back to rest. She covers a total distance of 1500 m in 270 s.

1. Sketch a speed-time graph to represent the athlete's run. [2]
2. Show that she accelerates for \(\frac{5V}{3}\) seconds. [2]
3. Show that \(V^2 - kV + 450 = 0\), where \(k\) is a constant to be found. [6]
4. Find the value of \(V\), justifying your answer. [4]

A train starts from rest at a station \(A\) and moves along a straight horizontal track. For the first 10 s, the train moves with constant acceleration 1.2 m s\(^{-2}\). For the next 24 s it moves at a constant acceleration 0.75 m s\(^{-2}\). It then moves with constant speed for \(T\) seconds. Finally it slows down with constant deceleration 3 m s\(^{-2}\) until it comes to a rest at station \(B\).

1. Show that, 34 s after leaving \(A\), the speed of the train is 30 m s\(^{-1}\). [3]
2. Sketch a speed-time graph to illustrate the motion of the train as it moves from \(A\) to \(B\). [3]
3. Find the distance moved by the train during the first 34 s of its journey from \(A\). [4]

The distance from \(A\) to \(B\) is 3 km.

1. Find the value of \(T\). [4]