3.02c Interpret kinematic graphs: gradient and area

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 moving in a straight line starts from rest at the point \(O\) at time \(t = 0\). At time \(t\) seconds, the velocity \(v\) m s\(^{-1}\) of the particle is given by $$v = 3t(t - 4), \quad 0 \leq t \leq 5,$$ $$v = 75t^{-1}, \quad 5 \leq t \leq 10.$$

1. Sketch a velocity-time graph for the particle for \(0 \leq t \leq 10\). [3]
2. Find the set of values of \(t\) for which the acceleration of the particle is positive. [2]
3. Show that the total distance travelled by the particle in the interval \(0 \leq t \leq 5\) is \(39\) m. [3]
4. Find, to \(3\) significant figures, the value of \(t\) at which the particle returns to \(O\). [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]

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]

A train starts from rest at a station \(S\) and accelerates at a constant rate for \(2x\) seconds to a speed of \(5x\) ms\(^{-1}\). It maintains this speed until 126 seconds after it left \(S\) and then decelerates at a constant rate until it comes to rest at another station \(T\), \(20x\) seconds after it left \(S\).

1. Sketch a velocity-time graph for this journey. [4 marks]

Given that the distance between \(S\) and \(T\) is \(5.4\) km,

1. show that \(x^2 + 7x = 120\). [4 marks]
2. Find the value of \(x\). [3 marks]

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]

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]

\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]

\includegraphics{figure_2} A particle starts from the point \(A\) and travels in a straight line. The diagram shows the \((t, v)\) graph, consisting of three straight line segments, for the motion of the particle during the interval \(0 \leq t \leq 290\).

1. Find the value of \(t\) for which the distance of the particle from \(A\) is greatest. [2]
2. Find the displacement of the particle from \(A\) when \(t = 290\). [3]
3. Find the total distance travelled by the particle during the interval \(0 \leq t \leq 290\). [2]

\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 \(0.02 \text{ m s}^{-2}\) for the first 40 s of the third stage, reaching a speed of \(V \text{ m s}^{-1}\). 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]

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]

A particle moves along the \(x\)-axis with velocity, \(v\) ms\(^{-1}\), at time \(t\) given by $$v = 24t - 6t^2.$$ The positive direction is in the sense of \(x\) increasing.

1. Find an expression for the acceleration of the particle at time \(t\). [2]
2. Find the times, \(t_1\) and \(t_2\), at which the particle has zero speed. [2]
3. Find the distance travelled between the times \(t_1\) and \(t_2\). [4]

Two girls, Marie and Nina, are members of an Olympic hockey team. They are doing fitness training. Marie runs along a straight line at a constant speed of \(6\) ms\(^{-1}\). Nina is stationary at a point O on the line until Marie passes her. Nina immediately runs after Marie until she catches up with her. The time, \(t\) s, is measured from the moment when Nina starts running. So when \(t = 0\), both girls are at O. Nina's acceleration, \(a\) ms\(^{-2}\), is given by \begin{align} a &= 4 - t \quad \text{for } 0 < t < 4,
a &= 0 \quad \text{for } t > 4. \end{align}

1. Show that Nina's speed, \(v\) ms\(^{-1}\), is given by \begin{align} v &= 4t - \frac{1}{2}t^2 \quad \text{for } 0 < t < 4,
   v &= 8 \quad \text{for } t > 4. \end{align} [3]
2. Find an expression for the distance Nina has run at time \(t\), for \(0 \leqslant t < 4\). Find how far Nina has run when \(t = 4\) and when \(t = 5\frac{1}{4}\). [4]
3. Show that Nina catches up with Marie when \(t = 5\frac{1}{4}\). [1]

The velocity, \(v\) ms\(^{-1}\), of a particle moving along a straight line is given by $$v = 3t^2 - 12t + 14,$$ where \(t\) is the time in seconds.

1. Find an expression for the acceleration of the particle at time \(t\). [2]
2. Find the displacement of the particle from its position when \(t = 1\) to its position when \(t = 3\). [4]
3. You are given that \(v\) is always positive. Explain how this tells you that the distance travelled by the particle between \(t = 1\) and \(t = 3\) has the same value as the displacement between these times. [2]

A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of \(15\text{ m s}^{-1}\). 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]

Fig. 1 is the velocity-time graph for the motion of a body. The velocity of the body is \(v\text{ m s}^{-1}\) at time \(t\) seconds. \includegraphics{figure_1} The displacement of the body from \(t = 0\) to \(t = 100\) is 1400 m. Find the value of \(V\). [4]

A particle travels in a straight line during the time interval \(0 \leqslant t \leqslant 12\), where \(t\) is the time in seconds. Fig. 1 is the velocity-time graph for the motion. \includegraphics{figure_3}

1. Calculate the acceleration of the particle in the interval \(0 < t < 6\). [2]
2. Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\). [2]
3. When \(t = 0\) the particle is at A. Calculate how close the particle gets to A during the interval \(4 \leqslant t \leqslant 12\). [2]

In this question take \(g\) as \(10\text{ m s}^{-2}\). A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6. \includegraphics{figure_4} For this model,

1. calculate the distance fallen from \(t = 0\) to \(t = 7\), [3]
2. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction, [3]
3. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\), [3]
4. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). [1]

The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = -\frac{3}{2}t^2 + \frac{19}{2}t + 7\).

1. Verify that \(v\) agrees with the values given in Fig. 6 at \(t = 2\), \(t = 6\) and \(t = 7\). [2]
2. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model. [7]

\includegraphics{figure_9} The diagram shows a velocity-time graph representing the motion of two cars \(A\) and \(B\) which are both travelling along a horizontal straight road. At time \(t = 0\), car \(B\), which is travelling with constant speed \(12 \mathrm{m s}^{-1}\), is overtaken by car \(A\) which has initial speed \(20 \mathrm{m s}^{-1}\). From \(t = 0\) car \(A\) travels with constant deceleration for 30 seconds. When \(t = 30\) the speed of car \(A\) is \(8 \mathrm{m s}^{-1}\) and the car maintains this speed in subsequent motion.

1. Calculate the deceleration of car \(A\). [2]
2. Determine the value of \(t\) when \(B\) overtakes \(A\). [4]

An object is moving in a straight line, with constant acceleration \(a\text{ m s}^{-2}\), over a time period of \(t\) seconds. It has an initial velocity \(u\) and final velocity \(v\) as shown in the graph below. \includegraphics{figure_13} Use the graph to show that $$v = u + at$$ [3 marks]