Q.The speed of a boat in still water is 14 km per hour. While going downstream it moves at the rate of 24 km per hour. Find the speed of the boat against the stream.

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.

• Shortest time: Point straight across. Time = $100 / 5 = 20$ s. In that 20 s, the current carries the boat $3 \times 20 = 60$ m downstream.
• Shortest path: You need $\sin\theta = 3/5$, so $\theta \approx 37^\circ$ upstream from the perpendicular. The net speed across is $5\cos\theta = 4$ m/s. Time = $100 / 4 = 25$ s.

The Key Insight

The boat's engine and the river current are independent. They don't fight each other — they add as vectors. The boat's path is the result of both, and you can choose to use the current (drift) or fight it (angle upstream), depending on what you want to optimize.

Once you see every problem as "what is $\vec{v}_{bw}$ and what is $\vec{v}_w$, and what am I trying to find?", the rest is just vector addition and a little trigonometry.