4 Water flows in a constant direction at a constant speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A boat travels in the water at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) relative to the water.
- The direction in which the boat travels relative to the water is perpendicular to the direction of motion of the water. The resultant velocity of the boat is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(74 ^ { \circ }\) to the direction of motion of the water, as shown in the diagram.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_120_164_662_488}
\captionsetup{labelformat=empty}
\caption{Velocity of the water}
\end{figure}
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{c220e6c4-2676-4022-8301-7d720dc082b2-3_126_186_667_890}
\captionsetup{labelformat=empty}
\caption{Velocity of the boat relative to the water}
\end{figure}
- Find \(V\).
- Show that \(u = 3.44\), correct to three significant figures.
- The boat changes course so that it travels relative to the water at an angle of \(45 ^ { \circ }\) to the direction of motion of the water. The resultant velocity of the boat is now of magnitude \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The velocity of the water is unchanged, as shown in the diagram below.
$$\xrightarrow { 3.44 \mathrm {~m} \mathrm {~s} ^ { - 1 } }$$
\includegraphics[max width=\textwidth, alt={}]{c220e6c4-2676-4022-8301-7d720dc082b2-3_132_273_1493_895}
Velocity of the boat relative to the water
\includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-3_232_355_1498_1384}
Find the value of \(v\).
(4 marks)