River crossing: perpendicular heading or minimum time (find drift and/or time)

5 A girl in a boat is rowing across a river, in which the water is flowing at \(0.1 \mathrm {~ms} ^ { - 1 }\). The velocity of the boat relative to the water is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is perpendicular to the bank, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{965a176a-848c-478d-a748-80fc9dfe4399-4_314_1152_468_450}

1. Find the magnitude of the resultant velocity of the boat.
2. Find the acute angle between the resultant velocity and the bank.
3. The width of the river is 15 metres.
   1. Find the time that it takes the boat to cross the river.
   2. Find the total distance travelled by the boat as it crosses the river.

4 A canoe is paddled across a river which has a width of 20 metres. The canoe moves from the point \(X\) on one bank of the river to the point \(Y\) on the other bank, so that its path is a straight line at an angle \(\alpha\) to the banks. The velocity of the canoe relative to the water is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) perpendicular to the banks. The water flows at \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) parallel to the banks. \includegraphics[max width=\textwidth, alt={}, center]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-10_469_1333_543_374} Model the canoe as a particle.

1. Find the magnitude of the resultant velocity of the canoe.
2. Find the angle \(\alpha\).
3. Find the time that it takes for the canoe to travel from \(X\) to \(Y\).
   \includegraphics[max width=\textwidth, alt={}]{f30b02da-a41e-44cb-b45f-9e6a3a9d0528-11_2486_1714_221_153}

4 A boat is crossing a river, which has two parallel banks. The width of the river is 20 metres. The water in the river is flowing at a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The boat sets off from the point \(O\) on one bank. The point \(A\) is directly opposite \(O\) on the other bank. The velocity of the boat relative to the water is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(70 ^ { \circ }\) to the bank. The boat lands at the point \(B\) which is 3 metres from \(A\). The angle between the actual path of the boat and the bank is \(\alpha ^ { \circ }\). The river and the velocities are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{788534a5-abbb-4d6a-87b2-c54e859a128a-10_490_1307_641_370}

1. Find the time that it takes for the boat to cross the river.
2. Find \(\alpha\).
3. \(\quad\) Find \(V\).

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.

1. 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]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_120_164_662_488} \captionsetup{labelformat=empty} \caption{Velocity of the water}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_126_186_667_890} \captionsetup{labelformat=empty} \caption{Velocity of the boat relative to the water}
   \end{figure}
   1. Find \(V\).
   2. Show that \(u = 3.44\), correct to three significant figures.
2. 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={}]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_132_273_1493_895}
   Velocity of the boat relative to the water \includegraphics[max width=\textwidth, alt={}, center]{6151e6ab-30af-4d1c-ab4a-e7dbad170cbf-004_232_355_1498_1384} Find the value of \(v\).
   (4 marks)

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.

1. 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}
   1. Find \(V\).
   2. Show that \(u = 3.44\), correct to three significant figures.
2. 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)

A river of width 40 m flows with uniform and constant speed between straight banks. A swimmer crosses as quickly as possible and takes 30 s to reach the other side. She is carried 25 m downstream. Find

1. the speed of the river, [2]
2. the speed of the swimmer relative to the water. [2]

\includegraphics{figure_1} A river is 50 m wide and flows between two straight parallel banks. The river flows with a uniform speed of \(\frac{2}{3}\) m s\(^{-1}\) parallel to the banks. The points \(A\) and \(B\) are on opposite banks of the river and \(AB\) is perpendicular to both banks of the river, as shown in Figure 1. Keith and Ian decide to swim across the river. The speed relative to the water of both swimmers is \(\frac{10}{9}\) m s\(^{-1}\). Keith sets out from \(A\) and crosses the river in the least possible time, reaching the opposite bank at the point \(C\). Find

1. the time taken by Keith to reach \(C\), [2]
2. the distance \(BC\). [2]

Ian sets out from \(A\) and swims in a straight line so as to land on the opposite bank at \(B\).

1. Find the time taken by Ian to reach \(B\). [4]