Understanding River Crossing Physics
When a swimmer attempts to cross a river, they are dealing with two distinct velocity vectors:
- Their own swimming velocity relative to the water ($\vec{v}_{sw}$).
- The velocity of the river current relative to the ground ($\vec{v}_r$).
To land at a point directly opposite the starting position, the swimmer's net velocity relative to the ground ($\vec{v}_g$) must be perpendicular to the riverbank.
The Mathematical Approach
Let the river flow in the horizontal direction (x-axis) and the straight path across be the vertical direction (y-axis). To reach the opposite point, the swimmer must point their body at an angle $\theta$ upstream against the current.
Mathematically, the horizontal component of the swimmer's velocity must exactly cancel out the river's velocity:
$$v_{sw} \cdot \sin(\theta) = v_r$$
Solving for the angle:
$$\theta = \arcsin\left(\frac{v_r}{v_{sw}}\right)$$
Why this works
By aiming upstream at this specific angle, the horizontal component of the swimmer's swimming velocity cancels the river flow, resulting in a resultant velocity vector that points straight across the river. Note that this is only possible if $v_{sw} > v_r$. If the river is faster than the swimmer, they will inevitably be carried downstream regardless of the angle chosen.