7 A car \(C\) is moving horizontally in a straight line with velocity \(v \mathrm {~ms} ^ { - 1 }\) at time \(t\) seconds, where \(v > 0\) and \(t \geqslant 0\). The acceleration, \(a \mathrm {~ms} ^ { - 2 }\), of \(C\) is modelled by the equation
\(a = v \left( \frac { 8 t } { 7 + 4 t ^ { 2 } } - \frac { 1 } { 2 } \right)\).
- In this question you must show detailed reasoning.
Find the times when the acceleration of \(C\) is zero.
At \(t = 0\) the velocity of \(C\) is \(17.5 \mathrm {~ms} ^ { - 1 }\) and at \(t = T\) the velocity of \(C\) is \(5 \mathrm {~ms} ^ { - 1 }\).
- By setting up and solving a differential equation, show that \(T\) satisfies the equation
$$T = 2 \ln \left( \frac { 7 + 4 T ^ { 2 } } { 2 } \right)$$
- Use an iterative formula, based on the equation in part (b), to find the value of \(T\), giving your answer correct to \(\mathbf { 4 }\) significant figures. Use an initial value of 11.25 and show the result of each step of the iteration process.
- The diagram below shows the velocity-time graph for the motion of \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-06_751_878_1372_322}
Find the time taken for \(C\) to decelerate from travelling at its maximum speed until it is travelling at \(5 \mathrm {~ms} ^ { - 1 }\).