Question

A boat is moving from the east bank to the west bank on a south flowing river. If the speed of the boat is 4 km/h and that of the river is 3 km/h. If the width of the river is 2 km, the distance travelled by the boat is

• A. 5 km
• B. 4 km
• C. 3 km
• D. 2.5 km
• E. 2 km

Hint:

The "distance travelled" in a river crossing problem refers to the actual path length over the ground, which is the hypotenuse of the width and the drift.

Answer 1

The Correct Option is D

Concept:
The boat's motion is the resultant of its own engine power (directed West) and the river's flow (directed South). Since these two velocities are perpendicular, the actual path is the diagonal (hypotenuse) of the velocity/displacement triangle.

Step 1: Calculate the time taken to cross the river.
The width of the river is crossed only by the boat's independent speed across the water: \[ t = \frac{\text{Width}}{\text{Boat Speed}} = \frac{2 \text{ km}}{4 \text{ km/h}} = 0.5 \text{ hours} \]

Step 2: Calculate the drift caused by the river.
While crossing, the boat is pushed downstream (South) by the river: \[ \text{Drift} = \text{River Speed} \times t = 3 \text{ km/h} \times 0.5 \text{ h} = 1.5 \text{ km} \]

Step 3: Determine total distance (Resultant Displacement).
Using the Pythagorean theorem for the right-angled triangle formed by the crossing width and the drift: \[ \text{Distance} = \sqrt{(\text{Width})^2 + (\text{Drift})^2} \] \[ \text{Distance} = \sqrt{2^2 + 1.5^2} = \sqrt{4 + 2.25} \] \[ \text{Distance} = \sqrt{6.25} = 2.5 \text{ km} \]