Question

The speed of a boat is 5 km/hr in still water. It crosses a river of width 1.0 m along the shortest possible path in 15 minute. The velocity of the river water (km/h) is
(1) 3
(2) 1
(3) 4
(4) 5

Answer 1

Question #60202, Physics / Mechanics | Relativity

The speed of a boat is $5\mathrm{km/hr}$ in still water. It crosses a river of width $1.0\mathrm{km}$ along the shortest possible path in 15 minute. The velocity of the river water $(\mathrm{km/h})$ is

(1) 3

(2) 1

(3) 4

(4) 5

Solution.

To cross the river in the shortest path, the boat must move perpendicular to the shores. Therefore, the boat should move at a specific angle $\alpha$, so that its own speed will compensate the lateral displacement caused by the flow speed:

The resulting speed of the boat in respect to the shores:

$$v _ {b o a t} ^ {2} = v _ {c r o s s} ^ {2} + v _ {f l o w} ^ {2}$$

At the same time:

$$\begin{array}{l} v _ {c r o s s} = \frac {d}{\Delta t} = \frac {1}{1 5 / 6 0} = 2. 5 \mathrm {k m / h} \\ v _ {f l o w} = \sqrt {v _ {b o a t} ^ {2} - v _ {c r o s s} ^ {2}} = \sqrt {5 ^ {2} - 2 . 5 ^ {2}} = 4. 3 3 \mathrm {k m / h} \\ \end{array}$$

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