3.02c Interpret kinematic graphs: gradient and area

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 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]

A graph indicating how the velocity, \(v\) m s\(^{-1}\), of a particle changes with respect to time, \(t\) seconds, is shown below. \includegraphics{figure_15}

1. Find the total distance travelled by the particle over the 8 second period shown. [3 marks]
2. A student claims that "The displacement of the particle is less than the distance travelled." State the range of values of \(t\) for which this claim is true. [1 mark]

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 particle is moving in a straight line with velocity \(v\text{ ms}^{-1}\) at time \(t\) seconds as shown by the graph below. \includegraphics{figure_15}

1. Use the trapezium rule with four strips to estimate the distance travelled by the particle during the time period \(20 \leq t \leq 100\) [4 marks]
2. Over the same time period, the curve can be very closely modelled by a particular quadratic. Explain how you could find an alternative estimate using this quadratic. [1 mark]

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 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]

A car is initially travelling with a constant velocity of \(15 \text{ m s}^{-1}\) for \(T\) s. It then decelerates at a constant rate for \(\frac{T}{2}\) s, reaching a velocity of \(10 \text{ m s}^{-1}\). It then immediately accelerates at a constant rate for \(\frac{3T}{2}\) s reaching a velocity of \(20 \text{ m s}^{-1}\).

1. Sketch a velocity–time graph to illustrate the motion. [3]
2. Given that the car travels a total distance of 1312.5 m over the journey described, find the value of \(T\). [4]

A racing car starts from rest at the point \(A\) and moves with constant acceleration of \(11 \text{ m s}^{-2}\) for \(8 \text{ s}\). The velocity it has reached after \(8 \text{ s}\) is then maintained for \(7 \text{ s}\). The racing car then decelerates from this velocity to \(40 \text{ m s}^{-1}\) in a further \(2 \text{ s}\), reaching point \(B\).

1. Sketch a velocity-time graph to illustrate the motion of the racing car. Include the top speed of the racing car in your sketch. [5]
2. Given that the distance between \(A\) and \(B\) is \(1404 \text{ m}\), find the value of \(T\). [3]

A jogger is running along a straight horizontal road. The jogger starts from rest and accelerates at a constant rate of \(0.4\,\text{m}\,\text{s}^{-2}\) until reaching a velocity of \(V\,\text{m}\,\text{s}^{-1}\). The jogger then runs at a constant velocity of \(V\,\text{m}\,\text{s}^{-1}\) before decelerating at a constant rate of \(0.08\,\text{m}\,\text{s}^{-2}\) back to rest. The jogger runs a total distance of \(880\,\text{m}\) in \(250\,\text{s}\).

1. Sketch the velocity-time graph for the jogger's journey. [2]
2. Show that \(3V^2 - 100V + 352 = 0\). [6]
3. Hence find the value of \(V\), giving a reason for your answer. [3]

The diagram below shows the velocity-time graph of a car moving along a straight road, where \(v\) m s\(^{-1}\) is the velocity of the car at time \(t\) s after it passes through the point \(A\). \includegraphics{figure_9}

1. Calculate the acceleration of the car at \(t = 6\). [2]
2. Jasmit says "The distance travelled by the car during the first 20 seconds of the car's motion is more than five times its displacement from \(A\) after the first 20 seconds of the car's motion". Give evidence to support Jasmit's statement. [3]

A particle moves along a straight line under the action of a variable force. The acceleration is given by $$a = \begin{cases} 30 - 6t, & \text{for } 0 \leqslant t \leqslant 10 \\ 6t - 90, & \text{for } 10 \leqslant t \leqslant 20 \end{cases}$$ where time \(t\) is measured in seconds and \(a\) in m s\(^{-2}\). The particle is at rest at the origin at \(t = 0\).

1. 1. Find the velocity \(v\) of the particle in terms of \(t\). Verify that \(v = 0\) when \(t = 10\) and \(t = 20\). [7]
   2. Sketch the velocity-time graph for the motion. [2]
2. Calculate the total distance travelled by the particle. [4]