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

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]

Answer all the questions. Two cyclists, \(A\) and \(B\), are cycling along the same straight horizontal track. The cyclists are modelled as particles and the motion of the cyclists is modelled as follows: • At time \(t = 0\), cyclist \(A\) passes through the point \(O\) with speed \(2\text{ms}^{-1}\) • Cyclist \(A\) is moving in a straight line with constant acceleration \(2\text{ms}^{-2}\) • At time \(t = 2\) seconds, cyclist \(B\) starts from rest at \(O\) • Cyclist \(B\) moves with constant acceleration \(6\text{ms}^{-2}\) along the same straight line and in the same direction as cyclist \(A\) • At time \(t = T\) seconds, \(B\) overtakes \(A\) at the point \(X\) Using the model,

1. sketch, on the same axes, for the interval from \(t = 0\) to \(t = T\) seconds, • a velocity-time graph for the motion of \(A\) • a velocity-time graph for the motion of \(B\) [2]
2. explain why the two graphs must cross before time \(t = T\) seconds, [1]
3. find the time when \(A\) and \(B\) are moving at the same speed, [2]
4. find the distance \(OX\) [5]

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]