# A river flows from south to north at 5 Meters/minute. A man on the West Bank of the river is...

## Question:

A river flows from south to north at 5 Meters/minute. A man on the West Bank of the river is capable of swimming at 10 meters/Minute in still water, wants to take the shortest time. In what direction he should move?

## Relative Velocity:

When an object is in motion along a moving surface, the relative velocity of the object depends on the location of the observer. It is important to make sure you are careful when deciding if the velocities would add together or subtract from each other.

Known Values

• Velocity of Man = {eq}v_m = 10\ \rm{m/min} {/eq}
• Velocity of River = {eq}v_r = 5\ \rm{m/mon} {/eq}

Required

• Time = {eq}t {/eq}
• Direction = {eq}\theta {/eq}

Calculations

We know that:

{eq}v = d/t {/eq}

Which shows that:

{eq}t = d/v {/eq}

We are looking for the smallest possible value of t

If we try to fight the river we will find:

{eq}t = d/vcos(\theta) {/eq}

This will give us the shortest path. But is there a faster one?

What if we let the river carry us?

{eq}t = d/vsin(\theta) {/eq}

However, sin(90) is just 1, so:

{eq}t = d/v {/eq}

We can see that this results in a shorter time!