A boat can travel at a speed of 3 m s^-1 on still water. A boatman wants to cross a river whilst covering the shortest possible distance. In what direction should he row with respect to the bank if the speed of the water is (i) 2 m s^-1, (ii) 4 m s^-1? Assume that the speed of the water is the

Question

A boat can travel at a speed of [latex] 3 m \ s^{-1} [/latex] on still water. A boatman wants to cross a river whilst covering the shortest possible distance. In what direction should he row with respect to the bank if the speed of the water is [latex] (i) 2 \ m \ s^{-1} [/latex], [latex] (ii) \ 4 \ m \ s^{-1} [/latex] ? Assume that the speed of the water is the same everywhere.

Accepted Answer

(i) The shortest path is one perpendicular to the bank and the boat goes in this direction if the boatman rows in the direction shown in Fig. S3.1.

The resultant speed of the boat (in the direction perpendicular to the bank) is [latex] \sqrt{5} \ m \ s^{-1} \approx 2.24 \ m \ s^{-1} [/latex]. The boatman has to row upstream at an angle [latex] \alpha [/latex] to the bank, where [latex] \alpha =\frac{2}{3} [/latex] ; this gives [latex] \alpha \approx 48^\circ [/latex] .

(ii) In this case, the current is so strong that the boat will move downstream even if the boatman rows at full speed against the stream. This means, in contrast to the previous case, he cannot choose his direction with respect to the bank and, in particular, he cannot travel across in a direction perpendicular to the bank.

The possible directions he can take may be determined by adding all the possible still-water velocities of the boat to the velocity of the river. Draw the velocity vector of the river and, from the endpoint of this vector, draw velocity vectors in all directions, with a magnitude equal to the speed of the boat in still water. The endpoints of these vectors will form a circle as shown in Fig. S3.2.

The possible resultant velocities of the boat can be obtained by joining the starting point of the velocity vector of the river to the points on this circle. The resultant corresponding to the shortest path will be the one that makes the greatest angle with the direction of the current, i.e. when the line of action of the resultant velocity vector is a tangent to the circle (see Fig. S3.3).
Thus, the velocity of the boat with respect to the shore is [latex] \sqrt{7} \ m \ s^{-1} \approx 2.65 \ m \ s^{-1} [/latex]. Again, the boatman has to row upstream, but this time at an angle [latex]\beta [/latex]to the bank, where [latex] cos \beta = \frac{3}{4} [/latex] yielding [latex] \beta \approx 41^\circ [/latex] . The figure also shows that in this case the distance travelled by the boat will be [latex] \frac{4}{3} [/latex] times the width of the river.

Question P.3: A boat can travel at a speed of 3 m s^-1 on still water. A b...

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