Swimming Problem - How can I do this?

Question

I have a problem with the following diagram. The triangle is just a possible path and may not be the shortest. If the boy has to get to the destination in the fastest way possible:

• What methods/models can I use to find out the shortest time? (Which includes swimming to each point of the point and continuing to walk to the destination)
• What logical assumptions are already made?
• What assumptions can I make?
• What are reasonable speeds of swimming and walking as well as distances?

I am utterly clueless so please provide some advice. Thx

Answer 1

My physics teacher asked similar question to me when I was introduced to kinematics. I hope it helps.

Dark line is Path Taken by Traveller.

Let $v_w$ , $v_s$ be velocities of walking and swimming respectively. and Assuming $v_s\lt v_w$

Let $$\eta=\frac{v_s}{v_w}$$ $$t_s=\frac{D}{v_s}$$

$$t_w=\frac{l-x}{v_w}$$ Total time taken is $$t(x)=t_s+t_w=\frac{D}{v_s}+\frac{l-x}{v_w}=\frac{l-x}{v_w}+\frac{\sqrt{d^2+x^2}}{v_s} $$ We need to Minimize $t(x)$ $$\frac{dt}{dx}=\frac{-1}{v_w}+\frac{2x}{2v_s\sqrt{d^2+x^2}}=0 $$

$$\Rightarrow xv_w=v_s\sqrt{d^2+x^2} $$

$$\Rightarrow x^2v_w^2=v_s^2(d^2+x^2) $$

$$\Rightarrow x^2v_w^2-x^2v_s^2=v_s^2d^2$$ $$\Rightarrow x=\frac{v_sd}{\sqrt{v_w^2-v_s^2}}=\frac{\eta d}{\sqrt{1-\eta^2}} $$

We can clearly see that $$\eta\lt1\Rightarrow v_s\lt v_w$$

Note : We can relate this to path of light when travelling through different mediums

where $\eta$ is ratio of velocities in different mediums ( Termed as Refractive Index ).

Answer 2

Presumably he walks on shore at some speed $w$ and swims at some speed $s$. The only paths to consider are walking some distance $d$ up the shore, then getting in the water and swimming a straight course to the destination. You need to set up an equation for the total time taken as a function of $d$ and minimize it by taking the derivative and setting to zero. So the time walking is $\frac dw$ Given $d$, how far does he swim? How long does it take? You need some dimensions from you problem that you didn't give. Now add the two times together and you have the total time as a function of $d$