Tim and John want to find the width of body of water.
They stand on opposite sides. John holds the mirror above his head. Tim holds a flashlight above his head and aims it at a spot between the water so that it reflects off into Tim's mirror. They know the following properties of light:
- 1) Light travels on the path that minimizes time
- 2) The shortest path between two points is a line that connects them
- 3) The speed of light = $v$
They know their heights, and Tim knows the angle he is aiming is flashlight at.
Find the width of the body of water.
Note: do not use angle of reflection = angle of incidence.
Hint: If you do not use calculus and all 3 properties, you are missing something.
Best Answer
Let $T$ be the point where Tim holds the flashlight. Let $J$ be John's mirror. Suppose the light reflects on the water at $P$. Let $T$ be a height $h$ above the ground and let $T′$ be the same distance below the ground (so that $TT'$ is perpendicular to the ground which it meets at $F$, its midpoint).
By (1) $TP+PJ$ must be minimised. But $TP=T′P$, so $T′P+PJ$ must be minimised and hence by 2) $P$ must lie on the straight line $T′J$.
[By definition $TF=FT′$ and $\angle PFT=\angle PFT′=90^\circ$. So the triangles $PFT,PFT′$ are congruent. So $TP=T′P$.]
So if Tim is aiming his flashlight at an angle $\alpha$ below the horizontal, we have that the width is $(h+h')$ cot $\alpha$, where $h'$ is the height of John's mirror above the ground.
The minimisation approach is known as Fermat's Principle. There is a good discussion of it in Feynman's Lectures on Physics.