The lifeguard problem: find the direction of sprint.

[tsuwal]

Homework Statement

This i the usual lifegard problem: we have a lifeguar that has a soil and water velocity, and wants to rescue a person at position (x,y). The lifeguard is ate the origin and th beach ends at position l, he wants to reach destination aa fast as possible.

We all know that this is accomplished by respecting the snell-descart law but what matters to the lifeguard is the angle he is going to run or the point in the waterline he is aiming, call it c.

Homework Equations

Snell-Descarte law

The Attempt at a Solution

I made the calculations but the expressian became really ugly, can this expression simplify?

 

[BruceW]

yeah, to be honest, I think a direct expression of C in terms of the other distances is going to be ugly, I don't think it will simplify into a nice expression.

I am trying to check your answer. So is v_p the velocity of the man on ground? And v_m the velocity of the man in water? I tried to work back from your answer, and I get to
\frac{x-c}{y-l} = \frac{y-l}{x-l} \frac{l}{c}
This doesn't make sense, so maybe it is worth checking through your working again?

 

[rude man]

Just a lot of yucky differentiation, right?

T1 = √[l² + c²]^1/2/v1
T2 = √[(y-l)² + (x-c)²]/v2
d(T1+T2)/dc = 0 etc.
Yuck!