Q.A person can row a boat 5 km an hour in still water. It takes him thrice as long to row upstream as to row downstream. Find the rate at which the stream is flowing.

Speed Current Boat Problems — A First Look

Imagine you're standing on the bank of a river that flows steadily downstream. You see a boat trying to cross. The boat's engine pushes it straight toward the opposite bank — but the river itself is moving sideways. What happens? The boat doesn't go straight across. It drifts downstream.

That's the core idea: the boat's motion is the sum of two independent motions — its own effort (relative to the water) and the water's flow (the current).

The Two Velocities

Every boat problem involves two distinct velocities:

1. Velocity of boat relative to water ($\vec{v}_{bw}$) — how fast the boat moves through still water, in whatever direction its engine points.
2. Velocity of water relative to ground ($\vec{v}_{w}$) — the speed and direction of the river current.

The boat's actual path over the ground is the vector sum:

This is not a guess — it's how relative motion works. The boat's engine moves it through the water, and the water itself carries everything along.

The Two Classic Problems

Type 1 — Crossing the river (shortest time)

You point the boat straight across, perpendicular to the current. The boat's own velocity is entirely across the river. The current pushes it downstream. The time to cross depends only on the boat's speed across and the river's width — the current does not affect how long it takes, only where it lands.

Time to cross:

If you point straight across ($\theta = 0$), then $t = \frac{d}{v_{bw}}$.

Type 2 — Crossing the river (shortest path)

You want to land exactly opposite your starting point. To cancel the drift, you must point the boat upstream at an angle, so that the upstream component of the boat's velocity exactly cancels the current. Then the net velocity is straight across.

Condition for no drift:

where $\theta$ is the angle the boat makes with the perpendicular to the bank.

A Concrete Example

A river 100 m wide flows at 3 m/s. A boat can do 5 m/s in still water. …