A driver is road-testing two minibuses, A and B, for a taxi company - AQA - A-Level Maths Pure - Question 15 - 2018 - Paper 2

For minibus B, the acceleration is:
$a_B = 0.024 t^3$
We integrate to find velocity:
$v_B = \int (0.024 t^3) dt = 0.006 t^4 + c$
As with A, since it starts from rest, ( c = 0 ):
$v_B = 0.006 t^4$
Now we integrate again for displacement:
$s_B = \int (0.006 t^4) dt = 0.0012 t^5 + k$
With ( k = 0 ) for initial position zero:
$s_B = 0.0012 t^5$
Set this equal to 100 metres to find time:
$0.0012 t^5 = 100$
This leads us to:

From the previous calculations, the time taken for A is approximately 9.657 seconds and for B is approximately 9.642 seconds. Thus, the company should purchase minibus B, as it travels 100 metres in the shortest time.

The models assume both minibuses start moving instantly when the flag is dropped. However, in practice, there could be reaction time for each driver before accelerating. If Driver B's reaction time is significantly longer than that of Driver A, A could travel 100 metres faster in reality, even though B has a calculated lower travel time.

Additionally, factors such as vehicle performance under different conditions, potential delays in starting, and driver efficiency may not be accurately reflected in the model used for the test.