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

For minibus B, we similarly begin with the acceleration function:

1. The acceleration of B is given as: $a = 0.024 t^3$

2. Integrate to find velocity (v): $v = \int a \, dt = \int 0.024 t^3 \, dt = 0.006 t^4 + c$ Starting from rest $v(0) = 0$ implies $c = 0$. So: $v = 0.006 t^4$

3. Integrate again to find displacement (s): $s = \int v \, dt = \int 0.006 t^4 \, dt = 0.0012 t^5 + k$ Again, using $s(0) = 0$ so $k = 0$ gives: $s = 0.0012 t^5$

4. Set displacement equal to 100 metres and solve for time: $0.0012 t^5 = 100$ $t^5 = \frac{100}{0.0012}$ $t = \sqrt[5]{83333.33333} \approx 9.642 \text{ seconds}$

5. The time taken for B to travel 100 metres is approximately 9.642 seconds.

Based on the computations:

• Time taken for minibus A: approximately 9.657 seconds
• Time taken for minibus B: approximately 9.642 seconds

Since minibus B takes less time to travel 100 metres, the company should decide to buy minibus B.

The assumption that both minibuses start moving immediately at $t = 0$ may not reflect real-world conditions. In practice, factors such as driver reaction time can significantly affect performance. If driver B's reaction time is greater than driver A's, this could result in minibus B taking longer to start moving, thus negating its theoretical advantage based solely on acceleration and time to travel 100 metres. Consequently, the company may choose a minibus based on flawed assumptions about immediate response times.