Understanding Relative Velocity in a Flowing River
When an object moves within a moving medium (like a swimmer in a river), the observed velocity depends on both the object's own effort and the medium's movement. This is a classic application of relative velocity in physics.
The Core Concepts
- Speed in Still Water ($v_s$): This is the speed the swimmer can achieve if there were no current (the river was stagnant).
- Velocity of the Stream ($v_c$): This is the speed at which the river water itself is moving.
- Downstream Speed ($v_d$): When the swimmer moves with the current, their speeds add up: $v_d = v_s + v_c$.
- Upstream Speed ($v_u$): When the swimmer moves against the current, the current opposes their motion: $v_u = v_s - v_c$.
Step-by-Step Solution
Given:
- Downstream speed ($v_d$) = $20 \text{ km/h}$
- Upstream speed ($v_u$) = $8 \text{ km/h}$
Equations:
- $v_s + v_c = 20$
- $v_s - v_c = 8$
Step 1: Calculate the swimmer's speed in still water ($v_s$) Add the two equations together: $(v_s + v_c) + (v_s - v_c) = 20 + 8$ $2v_s = 28$ $v_s = 14 \text{ km/h}$
Step 2: Calculate the velocity of the stream ($v_c$) Substitute $v_s$ back into the first equation: $14 + v_c = 20$ $v_c = 20 - 14$ $v_c = 6 \text{ km/h}$
Intuition
Think of the stream as a "moving sidewalk." If you walk on a moving sidewalk, your speed relative to the ground is either the sum of your walking speed and the sidewalk's speed (if walking in the same direction) or the difference (if walking against it). By measuring how fast you move relative to the ground in both directions, we can easily isolate the individual speeds of both you and the walkway.