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

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]

Fig. 4 shows the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the directions of the cartesian axes \(Ox\) and \(Oy\), respectively. O is the origin of the axes and of position vectors. \includegraphics{figure_1} The position vector of a particle is given by \(\mathbf{r} = 3t\mathbf{i} + (18t^2 - 11)\mathbf{j}\) for \(t \geq 0\), where \(t\) is time.

1. Show that the path of the particle cuts the \(x\)-axis just once. [2]
2. Find an expression for the velocity of the particle at time \(t\). Deduce that the particle never travels in the \(\mathbf{j}\) direction. [3]
3. Find the cartesian equation of the path of the particle, simplifying your answer. [3]

At time \(t\) seconds, a particle has position with respect to an origin O given by the vector $$\mathbf{r} = \begin{pmatrix} 8t \\ 10t^2 - 2t^3 \end{pmatrix},$$ where \(\begin{pmatrix} 1 \\ 0 \end{pmatrix}\) and \(\begin{pmatrix} 0 \\ 1 \end{pmatrix}\) are perpendicular unit vectors east and north respectively and distances are in metres.

1. When \(t = 1\), the particle is at P. Find the bearing of P from O. [2]
2. Find the velocity of the particle at time \(t\) and show that it is never zero. [3]
3. Determine the time(s), if any, when the acceleration of the particle is zero. [3]

A toy boat moves in a horizontal plane with position vector \(\mathbf{r} = x\mathbf{i} + y\mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are the standard unit vectors east and north respectively. The origin of the position vectors is at O. The displacements \(x\) and \(y\) are in metres. First consider only the motion of the boat parallel to the \(x\)-axis. For this motion $$x = 8t - 2t^2.$$ The velocity of the boat in the \(x\)-direction is \(v_x\) ms\(^{-1}\).

1. Find an expression in terms of \(t\) for \(v_x\) and determine when the boat instantaneously has zero speed in the \(x\)-direction. [3]

Now consider only the motion of the boat parallel to the \(y\)-axis. For this motion $$v_y = (t - 2)(3t - 2),$$ where \(v_y\) ms\(^{-1}\) is the velocity of the boat in the \(y\)-direction at time \(t\) seconds.

1. Given that \(y = 3\) when \(t = 1\), use integration to show that \(y = t^3 - 4t^2 + 4t + 2\). [4]

The position vector of the boat is given in terms of \(t\) by \(\mathbf{r} = (8t - 2t^2)\mathbf{i} + (t^3 - 4t^2 + 4t + 2)\mathbf{j}\).

1. Find the time(s) when the boat is due north of O and also the distance of the boat from O at any such times. [4]
2. Find the time(s) when the boat is instantaneously at rest. Find the distance of the boat from O at any such times. [5]
3. Plot a graph of the path of the boat for \(0 \leq t \leq 2\). [3]

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_6} The diagram shows part of the curve \(y = \frac{2x - 1}{(2x + 3)(x + 1)^2}\). Find the exact area of the shaded region, giving your answer in the form \(p + q \ln r\), where \(p\) and \(q\) are positive integers and \(r\) is a positive rational number. [10]

\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]

A vehicle, which begins at rest at point \(P\), is travelling in a straight line. For the first \(4\) seconds the vehicle moves with a constant acceleration of \(0.75\,\mathrm{m}\,\mathrm{s}^{-2}\) For the next \(5\) seconds the vehicle moves with a constant acceleration of \(-1.2\,\mathrm{m}\,\mathrm{s}^{-2}\) The vehicle then immediately stops accelerating, and travels a further \(33\,\mathrm{m}\) at constant speed.

1. Draw a velocity-time graph for this journey on the grid below. [3 marks] \includegraphics{figure_13}
2. Find the distance of the car from \(P\) after \(20\) seconds. [3 marks]

A car, initially at rest, is driven along a straight horizontal road. The graph below is a simple model of how the car's velocity, \(v\) metres per second, changes with respect to time, \(t\) seconds. \includegraphics{figure_13}

1. Find the displacement of the car when \(t = 45\) [3 marks]
2. Shona says: "This model is too simple. It is unrealistic to assume that the car will instantaneously change its acceleration." On the axes below sketch a graph, for the first 10 seconds of the journey, which would represent a more realistic model. [2 marks] \includegraphics{figure_13b}

A car, initially at rest, moves with constant acceleration along a straight horizontal road. One of the graphs below shows how the car's velocity, \(v\) m s\(^{-1}\), changes over time, \(t\) seconds. Identify the correct graph. Tick (✓) one box. [1 mark] \includegraphics{figure_11}

A particle is moving in a straight line such that its velocity, \(v \text{ m s}^{-1}\), changes with respect to time, \(t\) seconds, as shown in the graph below. \includegraphics{figure_15}

1. Show that the acceleration of the particle over the first 4 seconds is \(3.5 \text{ m s}^{-2}\) [1 mark]
2. The particle is initially at a fixed point \(P\) Show that the displacement of the particle from \(P\), when \(t = 9\), is 62 metres. [3 marks]

The graph shows how the speed of a cyclist varies during a timed section of length 120 metres along a straight track. \includegraphics{figure_15}

1. Find the acceleration of the cyclist during the first 10 seconds. [1 mark]
2. After the first 15 seconds, the cyclist travels at a constant speed of 5 m s⁻¹ for a further \(T\) seconds to complete the 120-metre section. Calculate the value of \(T\). [4 marks]

The graph below shows the velocity of an object moving in a straight line over a 20 second journey. \includegraphics{figure_4}

1. Find the maximum magnitude of the acceleration of the object. [1 mark]
2. The object is at its starting position at times 0, \(t_1\) and \(t_2\) seconds. Find \(t_1\) and \(t_2\) [4 marks]

The diagram below shows a velocity-time graph for a particle moving with velocity \(v \text{ m s}^{-1}\) at time \(t\) seconds. \includegraphics{figure_10} Which statement is correct? Tick (\(\checkmark\)) one box. [1 mark] The particle was stationary for \(9 \leq t \leq 12\) The particle was decelerating for \(12 \leq t \leq 20\) The particle had a displacement of zero when \(t = 6\) The particle's speed when \(t = 4\) was \(-12 \text{ m s}^{-1}\)

A car starting from rest moves forward in a straight line. The motion of the car is modelled by the velocity–time graph below: \includegraphics{figure_13} One of the following assumptions about the motion of the car is implied by the graph. Identify this assumption. [1 mark] Tick \((\checkmark)\) one box. The car never accelerates. The acceleration of the car is always positive. The acceleration of the car can change instantaneously. The acceleration of the car is never constant.

A particle moves on a straight line with a constant acceleration, \(a\) m s\(^{-2}\). The initial velocity of the particle is \(U\) m s\(^{-1}\). After \(T\) seconds the particle has velocity \(V\) m s\(^{-1}\). This information is shown on the velocity-time graph. \includegraphics{figure_12} The displacement, \(S\) metres, of the particle from its initial position at time \(T\) seconds is given by the formula $$S = \frac{1}{2}(U + V)T$$

1. By considering the gradient of the graph, or otherwise, write down a formula for \(a\) in terms of \(U\), \(V\) and \(T\). [1 mark]
2. Hence show that \(V^2 = U^2 + 2aS\) [3 marks]

The graph below models the velocity of a small train as it moves on a straight track for 20 seconds. The front of the train is at the point \(A\) when \(t = 0\) The mass of the train is 800kg. \includegraphics{figure_14}

1. Find the total distance travelled in the 20 seconds. [3 marks]
2. Find the distance of the front of the train from the point \(A\) at the end of the 20 seconds. [1 mark]
3. Find the maximum magnitude of the resultant force acting on the train. [2 marks]
4. Explain why, in reality, the graph may not be an accurate model of the motion of the train. [1 mark]

A particle P moves so that its position vector \(\mathbf{r}\) at time \(t\) is given by $$\mathbf{r} = (5 + 20t)\mathbf{i} + (95 + 10t - 5t^2)\mathbf{j}.$$

1. Determine the initial velocity of P. [3] At time \(t = T\), P is moving in a direction perpendicular to its initial direction of motion.
2. Determine the value of \(T\). [3]
3. Determine the distance of P from its initial position at time \(T\). [4]

A vehicle moves along a straight horizontal road. Points \(A\) and \(B\) lie on the road. As the vehicle passes point \(A\), it is moving with constant speed 15 ms\(^{-1}\). It travels with this constant speed for 2 minutes before a constant deceleration is applied for 12 seconds so that it comes to rest at point \(B\).

1. Find the distance \(AB\). [3]

The vehicle then reverses with a constant acceleration of 2 ms\(^{-2}\) for 8 seconds, followed by a constant deceleration of 1·6 ms\(^{-2}\), coming to rest at the point \(C\), which is between \(A\) and \(B\).

1. Calculate the time it takes for the vehicle to reverse from \(B\) to \(C\). [4]
2. Sketch a velocity-time graph for the motion of the vehicle. [3]
3. Determine the distance \(AC\). [2]

A car, starting from rest at a point \(A\), travels along a straight horizontal road towards a point \(B\). The distance between points \(A\) and \(B\) is 1·9 km. Initially, the car accelerates uniformly for 12 seconds until it reaches a speed of 26 ms\(^{-1}\). The car continues at 26 ms\(^{-1}\) for 1 minute, before decelerating at a constant rate of 0·75 ms\(^{-2}\) until it passes the point \(B\).

1. Show that the car travels 156 m while it is accelerating. [2]
2. 1. Work out the distance travelled by the car while travelling at a constant speed. [1]
   2. Hence calculate the length of time for which the car is decelerating until it passes the point \(B\). [5]
3. Sketch a displacement-time graph for the motion of the car between \(A\) and \(B\). [3]