6 In this question you should take \(\boldsymbol { g \) to be \(\mathbf { 1 0 } \mathrm { ms } ^ { \boldsymbol { - } \mathbf { 2 } }\).}
Piran finds a disused mineshaft on his land and wants to know its depth, \(d\) metres.
Local records state that the mineshaft is between 150 and 200 metres deep.
He drops a small stone down the mineshaft and records the time, \(T\) seconds, until he hears it hit the bottom. It takes 8.0 seconds.
Piran tries three models, \(\mathrm { A } , \mathrm { B }\) and C .
In model A, Piran uses the formula \(d = 5 T ^ { 2 }\) to estimate the depth.
- Find the depth that model A gives and comment on whether it is consistent with the local records.
Explain how the formula in model A is obtained.
In model B, Piran uses the speed-time graph in Fig. 6.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c8c96cf-5184-46e4-9c45-a8a80d0a6ff8-5_762_1176_1087_424}
\captionsetup{labelformat=empty}
\caption{Fig. 6}
\end{figure} - Calculate the depth of the mineshaft according to model B.
Comment on whether this depth is consistent with the local records.
- Describe briefly one respect in which model B is the same as model A and one respect in which it is different.
Piran then tries model C in which the speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), is given by
$$\begin{aligned}
& v = 10 t - t ^ { 2 } \text { for } 0 \leqslant t \leqslant 5 \\
& v = 25 \text { for } 5 < t \leqslant 8
\end{aligned}$$
- Calculate the depth of the mineshaft according to model C.
Comment on whether this depth is consistent with the local records.
- Describe briefly one respect in which model C is similar to model B and one respect in which it is different.