Understanding River Crossing Kinematics
River crossing problems are a classic application of relative velocity in two dimensions. To cross a river while compensating for a current, a swimmer must aim upstream so that the resultant velocity vector points directly across the river.
The Problem Statement
- Width of the river ($d$): $600\text{ m} = 0.6\text{ km}$
- Speed of swimmer in still water ($v_m$): $4\text{ km/h}$
- Speed of river current ($v_r$): $2\text{ km/h}$
Part 1: Finding the Direction
To reach a point directly opposite, the horizontal component of the swimmer's velocity must exactly cancel out the river's current. If $\theta$ is the angle with the perpendicular to the riverbank, the component of the swimmer's velocity against the current is $v_m \sin(\theta)$.
Setting these equal: $v_m \sin(\theta) = v_r$ $4 \sin(\theta) = 2$ $\sin(\theta) = 0.5$ $\theta = 30^\circ$
Conclusion: The man must swim at an angle of $30^\circ$ upstream from the line perpendicular to the river bank.
Part 2: Calculating the Time
The velocity of the swimmer relative to the ground ($v_g$) is the vertical component: $v_g = v_m \cos(\theta) = 4 \cos(30^\circ) = 4 \times \frac{\sqrt{3}}{2} = 2\sqrt{3} \approx 3.464\text{ km/h}$
Now, calculate the time ($t$): $t = \frac{d}{v_g} = \frac{0.6}{2\sqrt{3}} \approx 0.1732\text{ hours}$ To convert to minutes: $0.1732 \times 60 \approx 10.39\text{ minutes}$
Result: He will reach the point in approximately $10.4$ minutes.