Q.A man can swim at 4 km/h in still water; the river flows at 3 km/h. He wishes to cross the river of width 400 m by the shortest path (arriving directly opposite his starting point). At what angle to the straight-across direction should he head, and how long will he take to cross?

Relative Velocity in Two Dimensions

You already know relative velocity in one dimension — the speed of a car measured from another moving car. Two dimensions is the same idea, but now velocities have both magnitude and direction, so subtraction becomes vector subtraction.

The Intuition

Imagine you're standing on a train platform. A train passes you at 15 m/s. From your frame, the train's velocity is 15 m/s forward. Now imagine you're on that train, walking toward the front at 2 m/s relative to the train. Your velocity relative to the platform is 15 + 2 = 17 m/s forward. That's one-dimensional relative velocity: just add or subtract numbers.

Now picture a boat crossing a river. The boat points straight across, but the river flows sideways. From the bank, the boat moves diagonally — its velocity is the vector sum of its own velocity (relative to water) and the water's velocity (relative to bank). To find how the boat moves relative to the bank, you add vectors.

The reverse problem: you know the boat's velocity relative to bank and the water's velocity. To find the boat's velocity relative to water, you subtract vectors. That subtraction is the core of relative velocity in two dimensions.

The Precise Statement

If object A has velocity $\vec{v}_A$ and object B has velocity $\vec{v}_B$, both measured in the same frame (say, the ground), then the velocity of A relative to B is:

This is a vector equation. It says: to find how A appears to move from B's perspective, take A's velocity and subtract B's velocity (vector subtraction). Equivalently, $\vec{v}_{A/B}$ is the velocity you would measure if you were riding on B.

Why Vector Subtraction?

In one dimension, subtraction is just flipping a sign. In two dimensions, velocities have components. Subtracting vectors means subtracting corresponding components:

If $\vec{v}_A = v_{Ax}\,\hat{i} + v_{Ay}\,\hat{j}$ and $\vec{v}_B = v_{Bx}\,\hat{i} + v_{By}\,\hat{j}$, then

Geometrically, you place the tails of $\vec{v}_A$ and $\vec{v}_B$ together. The vector from the tip of $\vec{v}_B$ to the tip of $\vec{v}_A$ is $\vec{v}_{A/B}$.

The Two Classic Problems

Rain-Man Problem: A man walks in rain. Rain falls vertically downward at speed $v_r$ relative to ground. The man walks horizontally at speed $v_m$. To him, the rain appears to come at an angle. Why? Because the rain's velocity relative to the man is $\vec{v}_{r/m} = \vec{v}_r - \vec{v}_m$. Since $\vec{v}_r$ is downward and $\vec{v}_m$ is horizontal, the relative velocity is diagonal — the man must tilt his umbrella forward. …