14 The velocity of a car, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds, is being modelled. Initially the car has velocity \(5 \mathrm {~ms} ^ { - 1 }\) and it accelerates to \(11.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 4 seconds.
In model A, the acceleration is assumed to be uniform.
- Find an expression for the velocity of the car at time \(t\) using this model.
- Explain why this model is not appropriate in the long term.
Model A is refined so that the velocity remains constant once the car reaches \(17.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
- Sketch a velocity-time graph for the motion of the car, making clear the time at which the acceleration changes.
- Calculate the displacement of the car in the first 20 seconds according to this refined model.
In model B, the velocity of the car is given by
$$v = \begin{cases} 5 + 0.6 t ^ { 2 } - 0.05 t ^ { 3 } & \text { for } 0 \leqslant t \leqslant 8
17.8 & \text { for } 8 < t \leqslant 20 \end{cases}$$ - Show that this model gives an appropriate value for \(v\) when \(t = 4\).
- Explain why the value of the acceleration immediately before the velocity becomes constant is likely to mean that model B is a better model than model A.
- Show that model B gives the same value as model A for the displacement at time 20 s .