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

\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]

An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of 8 m s\(^{-1}\). This speed is then maintained for \(T\) seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s.

1. In the space below, sketch a speed-time graph to illustrate the motion of the athlete. [3]
2. Calculate the value of \(T\). [5]

\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]

A man travels in a lift to the top of a tall office block. The lift starts from rest on the ground floor and moves vertically. It comes to rest again at the top floor, having moved a vertical distance of \(27 \text{ m}\). The lift initially accelerates with a constant acceleration of \(2 \text{ m s}^{-1}\) until it reaches a speed of \(3 \text{ m s}^{-1}\). It then moves with a constant speed of \(3 \text{ m s}^{-1}\) for \(T\) seconds. Finally it decelerates with a constant deceleration for \(2.5 \text{ s}\) before coming to rest at the top floor.

1. Sketch a speed-time graph for the motion of the lift. [2]
2. Hence, or otherwise, find the value of \(T\). [3]
3. Sketch an acceleration-time graph for the motion of the lift. [3]

The mass of the man is \(80 \text{ kg}\) and the mass of the lift is \(120 \text{ kg}\). The lift is pulled up by means of a vertical cable attached to the top of the lift. By modelling the cable as light and inextensible, find

1. the tension in the cable when the lift is accelerating, [3]
2. the magnitude of the force exerted by the lift on the man during the last \(2.5 \text{ s}\) of the motion. [3]

A train is travelling at \(10 \text{ m s}^{-1}\) on a straight horizontal track. The driver sees a red signal 135 m ahead and immediately applies the brakes. The train immediately decelerates with constant deceleration for 12 s, reducing its speed to \(3 \text{ m s}^{-1}\). The driver then releases the brakes and allows the train to travel at a constant speed of \(3 \text{ m s}^{-1}\) for a further 15 s. He then applies the brakes again and the train slows down with constant deceleration, coming to rest as it reaches the signal.

1. Sketch a speed-time graph to show the motion of the train, [3]
2. Find the distance travelled by the train from the moment when the brakes are first applied to the moment when its speed first reaches \(3 \text{ m s}^{-1}\). [2]
3. Find the total time from the moment when the brakes are first applied to the moment when the train comes to rest. [5]

Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed 25 m s\(^{-1}\). At time \(t = 0\), \(P\) overtakes \(Q\) which is moving with constant speed 20 m s\(^{-1}\). From \(t = T\) seconds, \(P\) decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25\) s, \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).

1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\). [4]
2. Find the value of \(T\). [8]

A girl runs a 400 m race in a time of 84 s. In a model of this race, it is assumed that, starting from rest, she moves with constant acceleration for 4 s, reaching a speed of 5 m s\(^{-1}\). She maintains this speed for 60 s and then moves with constant deceleration for 20 s, crossing the finishing line with a speed of \(V\) m s\(^{-1}\).

1. Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race. [2]
2. Find the distance run by the girl in the first 64 s of the race. [3]
3. Find the value of \(V\). [5]
4. Find the deceleration of the girl in the final 20 s of her race. [2]

A car is travelling along a straight horizontal road. The car takes 120 s to travel between two sets of traffic lights which are 2145 m apart. The car starts from rest at the first set of traffic lights and moves with constant acceleration for 30 s until its speed is \(22 \text{ m s}^{-1}\). The car maintains this speed for \(T\) seconds. The car then moves with constant deceleration, coming to rest at the second set of traffic lights.

1. Sketch, in the space below, a speed-time graph for the motion of the car between the two sets of traffic lights. [2]
2. Find the value of \(T\). [3]

A motorcycle leaves the first set of traffic lights 10 s after the car has left the first set of traffic lights. The motorcycle moves from rest with constant acceleration, \(a \text{ m s}^{-2}\), and passes the car at the point \(A\) which is 990 m from the first set of traffic lights. When the motorcycle passes the car, the car is moving with speed \(22 \text{ m s}^{-1}\).

1. Find the time it takes for the motorcycle to move from the first set of traffic lights to the point \(A\). [4]
2. Find the value of \(a\). [2]

A racing car is travelling on a straight horizontal road. Its initial speed is \(25\) m s\(^{-1}\) and it accelerates for \(4\) s to reach a speed of \(V\) m s\(^{-1}\). It then travels at a constant speed of \(V\) m s\(^{-1}\) for a further \(8\) s. The total distance travelled by the car during this \(12\) s period is \(600\) m.

1. Sketch a speed-time graph to illustrate the motion of the car during this \(12\) s period. [2]
2. Find the value of \(V\). [4]
3. Find the acceleration of the car during the initial \(4\) s period. [2]

A car starts from rest at a point \(S\) on a straight racetrack. The car moves with constant acceleration for 20 s, reaching a speed of 25 m s\(^{-1}\). The car then travels at a constant speed of 25 m s\(^{-1}\) for 120 s. Finally it moves with constant deceleration, coming to rest at a point \(F\).

1. In the space below, sketch a speed-time graph to illustrate the motion of the car. [2]

The distance between \(S\) and \(F\) is 4 km.

1. Calculate the total time the car takes to travel from \(S\) to \(F\). [3]

A motorcycle starts at \(S\), 10 s after the car has left \(S\). The motorcycle moves with constant acceleration from rest and passes the car at a point \(P\) which is 1.5 km from \(S\). When the motorcycle passes the car, the motorcycle is still accelerating and the car is moving at a constant speed. Calculate

1. the time the motorcycle takes to travel from \(S\) to \(P\), [5]
2. the speed of the motorcycle at \(P\). [2]

A man is driving a car on a straight horizontal road. He sees a junction \(S\) ahead, at which he must stop. When the car is at the point \(P\), 300 m from \(S\), its speed is \(30 \text{ m s}^{-1}\). The car continues at this constant speed for 2 s after passing \(P\). The man then applies the brakes so that the car has constant deceleration and comes to rest at \(S\).

1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car in moving from \(P\) to \(S\). [2]
2. Find the time taken by the car to travel from \(P\) to \(S\). [3]

A train \(T_1\) moves from rest at Station \(A\) with constant acceleration \(2 \text{ m s}^{-2}\) until it reaches a speed of \(36 \text{ m s}^{-1}\). In maintains this constant speed for 90 s before the brakes are applied, which produce constant retardation \(3 \text{ m s}^{-2}\). The train \(T_1\) comes to rest at station \(B\).

1. Sketch a speed-time graph to illustrate the journey of \(T_1\) from \(A\) to \(B\). [3]
2. Show that the distance between \(A\) and \(B\) is 3780 m. [5]

\includegraphics{figure_3} A second train \(T_2\) takes 150 s to move form rest at \(A\) to rest at \(B\). Figure 3 shows the speed-time graph illustrating this journey.

1. Explain briefly one way in which \(T_1\)'s journey differs from \(T_2\)'s journey. [1]
2. Find the greatest speed, in m s\(^{-1}\), attained by \(T_2\) during its journey. [3]

A particle \(P\) of mass \(0.3\) kg is moving under the action of a single force \(F\) newtons. At time \(t\) seconds the velocity of \(P\), v m s\(^{-1}\), is given by $$\mathbf{v} = 3t^2\mathbf{i} + (6t - 4)\mathbf{j}.$$

1. Calculate, to 3 significant figures, the magnitude of \(\mathbf{F}\) when \(t = 2\). [5]

When \(t = 0\), \(P\) is at the point \(A\). The position vector of \(A\) with respect to a fixed origin \(O\) is \((3\mathbf{i} - 4\mathbf{j})\) m. When \(t = 4\), \(P\) is at the point \(B\).

1. Find the position vector of \(B\). [5]

The velocity v m s\(^{-1}\) of a particle \(P\) at time \(t\) seconds is given by $$\mathbf{v} = (3t - 2)\mathbf{i} - 5t\mathbf{j}.$$

1. Show that the acceleration of \(P\) is constant. [2]

At \(t = 0\), the position vector of \(P\) relative to a fixed origin \(O\) is 3i m.

1. Find the distance of \(P\) from \(O\) when \(t = 2\). [6]

A particle \(P\) of mass 0.3 kg is moving under the action of a single force \(\mathbf{F}\) newtons. At time \(t\) seconds the velocity of \(P\), \(\mathbf{v}\) m s\(^{-1}\), is given by $$\mathbf{v} = 3t\mathbf{i} + (6t - 4)\mathbf{j}.$$

1. Calculate, to 3 significant figures, the magnitude of \(\mathbf{F}\) when \(t = 2\). [5]

When \(t = 0\), \(P\) is at the point \(A\). The position vector of \(A\) with respect to a fixed origin \(O\) is \((3\mathbf{i} - 4\mathbf{j})\) m. When \(t = 4\), \(P\) is at the point \(B\).

1. Find the position vector of \(B\). [5]

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]

A particle \(P\) moves in a straight line such that its displacement from a fixed point \(O\) at time \(t\) s is \(y\) metres. The graph of \(y\) against \(t\) is as shown.

1. Write down the velocity of \(P\) when
   1. \(t = 1\), \quad (ii) \(t = 10\). \hfill [2 marks]
2. State the total distance travelled by \(P\). \hfill [2 marks]
3. Write down a formula for \(y\) in terms of \(t\) when \(2 \leq t < 4\). \hfill [3 marks]
4. Sketch a velocity-time graph for the motion of \(P\) during the twelve seconds. \hfill [3 marks]
5. Find the maximum speed of \(P\) during the motion. \hfill [3 marks]

\includegraphics{figure_7} A car \(P\) starts from rest and travels along a straight road for \(600\) s. The \((t, v)\) graph for the journey is shown in the diagram. This graph consists of three straight line segments. Find

1. the distance travelled by \(P\), [3]
2. the deceleration of \(P\) during the interval \(500 < t < 600\). [2]

Another car \(Q\) starts from rest at the same instant as \(P\) and travels in the same direction along the same road for \(600\) s. At time \(t\) s after starting the velocity of \(Q\) is \((600t^2 - t^3) \times 10^{-6}\) m s\(^{-1}\).

1. Find an expression in terms of \(t\) for the acceleration of \(Q\). [2]
2. Find how much less \(Q\)'s deceleration is than \(P\)'s when \(t = 550\). [2]
3. Show that \(Q\) has its maximum velocity when \(t = 400\). [2]
4. Find how much further \(Q\) has travelled than \(P\) when \(t = 400\). [6]

A man drives a car on a horizontal straight road. At \(t = 0\), where the time \(t\) is in seconds, the car runs out of petrol. At this instant the car is moving at \(12\) m s\(^{-1}\). The car decelerates uniformly, coming to rest when \(t = 8\). The man then walks back along the road at \(0.7\) m s\(^{-1}\) until he reaches a petrol station a distance of \(420\) m from his car. After his arrival at the petrol station it takes him \(250\) s to obtain a can of petrol. He is then given a lift back to his car on a motorcycle. The motorcycle starts from rest and accelerates uniformly until its speed is \(20\) m s\(^{-1}\); it then decelerates uniformly, coming to rest at the stationary car at time \(t = T\).

1. Sketch the shape of the \((t, v)\) graph for the man for \(0 \leq t \leq T\). [Your sketch need not be drawn to scale; numerical values need not be shown.] [5]
2. Find the deceleration of the car for \(0 < t < 8\). [2]
3. Find the value of \(T\). [4]

A man travels \(360\) m along a straight road. He walks for the first \(120\) m at \(1.5\) m s\(^{-1}\), runs the next \(180\) m at \(4.5\) m s\(^{-1}\), and then walks the final \(60\) m at \(1.5\) m s\(^{-1}\). The man's displacement from his starting point after \(t\) seconds is \(x\) metres.

1. Sketch the \((t, x)\) graph for the journey, showing the values of \(t\) for which \(x = 120, 300\) and \(360\). [5]

A woman jogs the same \(360\) m route at constant speed, starting at the same instant as the man and finishing at the same instant as the man.

1. Draw a dotted line on your \((t, x)\) graph to represent the woman's journey. [1]
2. Calculate the value of \(t\) at which the man overtakes the woman. [5]

The driver of a car accelerating uniformly from rest sees an obstruction. She brakes immediately bringing the car to rest with constant deceleration at a distance of \(6\) m from its starting point. The car travels in a straight line and is in motion for \(3\) seconds.

1. Sketch the \((t, v)\) graph for the car's motion. [2]
2. Calculate the maximum speed of the car during its motion. [3]
3. Hence, given that the acceleration of the car is \(2.4\) m s\(^{-2}\), calculate its deceleration. [4]

\includegraphics{figure_7} The diagram shows the \((t, v)\) graphs for two particles \(A\) and \(B\) which move on the same straight line. The units of \(v\) and \(t\) are \(\text{m s}^{-1}\) and \(\text{s}\) respectively. Both particles are at the point \(S\) on the line when \(t = 0\). The particle \(A\) is initially at rest, and moves with acceleration \(0.18t\text{ m s}^{-2}\) until the two particles collide when \(t = 16\). The initial velocity of \(B\) is \(U\text{ m s}^{-1}\) and \(B\) has variable acceleration for the first five seconds of its motion. For the next ten seconds of its motion \(B\) has a constant velocity of \(9\text{ m s}^{-1}\); finally \(B\) moves with constant deceleration for one second before it collides with \(A\).

1. Calculate the value of \(t\) at which the two particles have the same velocity. [4]

For \(0 \leq t \leq 5\) the distance of \(B\) from \(S\) is \((Ut + 0.08t^2)\text{ m}\).

1. Calculate \(U\) and verify that when \(t = 5\), \(B\) is \(25\text{ m}\) from \(S\). [4]
2. Calculate the velocity of \(B\) when \(t = 16\). [5]

\includegraphics{figure_3} A woman runs from \(A\) to \(B\), then from \(B\) to \(A\) and then from \(A\) to \(B\) again, on a straight track, taking 90 s. The woman runs at a constant speed throughout. Fig. 1 shows the \((t, v)\) graph for the woman.

1. Find the total distance run by the woman. [3]
2. Find the distance of the woman from \(A\) when \(t = 50\) and when \(t = 80\), [3]

\includegraphics{figure_4} At time \(t = 0\), a child also starts to move, from \(A\), along \(AB\). The child walks at a constant speed for the first 50 s and then at an increasing speed for the next 40 s. Fig. 2 shows the \((t, v)\) graph for the child; it consists of two straight line segments.

1. At time \(t = 50\), the woman and the child pass each other, moving in opposite directions. Find the speed of the child during the first 50 s. [3]
2. At time \(t = 80\), the woman overtakes the child. Find the speed of the child at this instant. [3]

A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of 15 m s\(^{-1}\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.

1. Sketch a velocity-time graph for the motion. [3]
2. Calculate the distance travelled by the cyclist. [3]

The displacement, \(x\) m, from the origin O of a particle on the \(x\)-axis is given by $$x = 10 + 36t + 3t^2 - 2t^3,$$ where \(t\) is the time in seconds and \(-4 \leqslant t \leqslant 6\).

1. Write down the displacement of the particle when \(t = 0\). [1]
2. Find an expression in terms of \(t\) for the velocity, \(v\) ms\(^{-1}\), of the particle. [2]
3. Find an expression in terms of \(t\) for the acceleration of the particle. [2]
4. Find the maximum value of \(v\) in the interval \(-4 \leqslant t \leqslant 6\). [3]
5. Show that \(v = 0\) only when \(t = -2\) and when \(t = 3\). Find the values of \(x\) at these times. [5]
6. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\). [3]
7. Determine how many times the particle passes through O in the interval \(-4 \leqslant t \leqslant 6\). [3]