Model refinement or criticism

11 A sports car accelerates along a straight road from rest. After 5 s its velocity is \(9 \mathrm {~ms} ^ { - 1 }\). In model A, the acceleration is assumed to be constant.

1. Calculate the distance travelled by the car in the first 5 seconds according to model A . In model B , the velocity \(v\) in \(\mathrm { ms } ^ { - 1 }\) is given by \(\mathrm { v } = 0.05 \mathrm { t } ^ { 3 } + \mathrm { kt }\), where \(t\) is the time in seconds after the start and \(k\) is a constant.
2. Find the value of \(k\) which gives the correct value of \(v\) when \(t = 5\).
3. Using this value of \(k\) in model B , calculate the acceleration of the car when \(t = 5\). The car travels 16 m in the first 5 seconds.
4. Show that model B, with the value of \(k\) found in part (b), better fits this information than model A does.

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.

1. Find an expression for the velocity of the car at time \(t\) using this model.
2. 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 \(17.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Sketch a velocity-time graph for the motion of the car, making clear the time at which the acceleration changes.
4. 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 $$v = \begin{cases} 5 + 0.6 t ^ { 2 } - 0.05 t ^ { 3 } & \text { for } 0 \leqslant t \leqslant 8 \\ 17.8 & \text { for } 8 < t \leqslant 20 \end{cases}$$
5. Show that this model gives an appropriate value for \(v\) when \(t = 4\).
6. 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.
7. Show that model B gives the same value as model A for the displacement at time 20 s .

9 A particle moves along a straight line such that its displacement from \(O\), a fixed point on the line, is \(x\). The particle travels from rest from the point \(P\), where \(x = 2\), to the point \(Q\), where \(x = 5.6\). All distances are in metres. Two models for the motion of the particle are proposed.

1. In Model 1, the acceleration of the particle is assumed to be constant and the particle takes 18 seconds to travel from \(P\) to \(Q\). Find the velocity of the particle when it reaches \(Q\).
2. In Model 2, the velocity after \(t\) seconds is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = \frac { 1 } { 270 } \left( 18 t - t ^ { 2 } \right)\).
   1. Write down the values of \(t\) when \(v = 0\).
   2. Show that \(x = 5.6\) when \(t = 18\).
   3. The particle represents a fragile instrument that is being moved from \(P\) to \(Q\) across a laboratory. Explain why Model 2 might be more appropriate than Model 1.

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}