5 Jack and Jemima are pulling a boat along a straight level canal.
The resistance to the motion of the boat is modelled as constant and equal to 1200 N .
Jack and Jemima walk in the same direction on paths on opposite sides of the canal. They each walk forwards at the same steady speed, keeping level with each other so that the distance between them is always 6 m . Jack and Jemima each pull a long light inextensible rope attached to the boat; initially they hold their ropes so the distance from each of them to the boat is 5 m , as shown in Fig. 5.1.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b808042-95b8-4862-8355-3979c1981089-4_417_1109_605_246}
\captionsetup{labelformat=empty}
\caption{Fig. 5.1}
\end{figure}
- Explain why the tension will be the same in each rope.
- Find the tension in each rope.
Jemima then gradually releases more rope, so that the distance between her and the boat is 7 m . Jack and Jemima continue to walk at the same steady speed along the paths, but the position of the boat changes so that Jemima's rope makes an angle of \(\theta\) with the path and Jack's rope makes an angle of \(\phi\) with the path, as shown in Fig. 5.2.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3b808042-95b8-4862-8355-3979c1981089-4_513_1109_1610_246}
\captionsetup{labelformat=empty}
\caption{Fig. 5.2}
\end{figure} - - Show that \(\sin \phi = \frac { 1 } { 5 }\).
- Show that \(\sin \theta = \frac { 5 } { 7 }\).
- Find the tension in each rope in this new equilibrium position.
- Without further calculation, state the effect on the tensions in the ropes if Jack now lengthens his rope to 7 m , the same length as Jemima's rope.
- Suggest how the modelling assumption used in this question could be improved.