2 The speed of a 100 metre runner in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) is measured electronically every 4 seconds.
The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time.
Fig. 6 also illustrates Model P in which the points are joined by straight lines.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4f80ea36-001f-4a00-849f-542f5072516b-2_1022_1503_524_290}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{figure}
- Use Model P to estimate
(A) the distance the runner has gone at the end of 12 seconds,
(B) how long the runner took to complete 100 m .
A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by
$$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 } .$$ - Verify that Model Q gives the correct speed for \(t = 8\).
- Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
- The runner was timed at 11.35 seconds for the 100 m .
Which model places the runner closer to the finishing line at this time?
In this question take \(g\) as \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4f80ea36-001f-4a00-849f-542f5072516b-3_658_1101_281_503}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{figure}
For this model, - calculate the distance fallen from \(t = 0\) to \(t = 7\),
- find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction,
- obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\),
- state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\).
The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = - \frac { 3 } { 2 } t ^ { 2 } + \frac { 19 } { 2 } t + 7\).
- Verify that \(v\) agrees with the values given in Fig, 6 at \(t = 2 , t = 6\) and \(t = 7\).
- Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model.