6 A toy car is travelling in a straight horizontal line.
One model of the motion for \(0 \leqslant t \leqslant 8\), where \(t\) is the time in seconds, is shown in the velocity-time graph Fig. 6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4957086c-fd1c-4cdc-bbdb-1959b3b21b2d-4_474_1196_580_424}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{figure}
- Calculate the distance travelled by the car from \(t = 0\) to \(t = 8\).
- How much less time would the car have taken to travel this distance if it had maintained its initial speed throughout?
- What is the acceleration of the car when \(t = 1\) ?
From \(t = 8\) to \(t = 14\), the car travels 58.5 m with a new constant acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
- Find \(a\).
A second model for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the toy car is
$$v = 12 - 10 t + \frac { 9 } { 4 } t ^ { 2 } - \frac { 1 } { 8 } t ^ { 3 } , \text { for } 0 \leqslant t \leqslant 8$$
This model agrees with the values for \(v\) given in Fig. 6 for \(t = 0,2,4\) and 6. [Note that you are not required to verify this.] Use this second model to answer the following questions.
- Calculate the acceleration of the car when \(t = 1\).
- Initially the car is at A. Find an expression in terms of \(t\) for the displacement of the car from A after the first \(t\) seconds of its motion.
Hence find the displacement of the car from A when \(t = 8\).
- Explain with a reason what this model predicts for the motion of the car between \(t = 2\) and \(t = 4\).