Ulf the glider pilot.

We are asked how high Ulf must go in the thermal to reach Munich in minimum time. The diagram looks like this:

The desired answer is the height *h*. We know the distance *D* and the speed *v*_{1} at which Ulf travels from S to T in the thermal. His speed from T to M depends on the angle marked in red, called a
.

*v _{a} = v_{o}*tana
=

Our concepts are linear motion at constant speed, and the use of calculus to minimize a function. We need to find the time *t* as a function of our variable *h*.

Then the time to reach Munich (M) is the time to go from S to T plus the time to go from T to M:

*t = h/ v*_{1} + *L/ v _{a}*

Now we make use of the hint: if the angle a
is very small, then sina
»
tana
and that means *L »
D. *Then the function we want is:

*t = h/ v*_{1} + *D/( v _{o}h/D)*

*= h/ v*_{1} + *D ^{2}/v_{o}h*

Now we differentiate:

*dt/dh = 1/ v*_{1} - *D ^{2}/v_{o}h^{2}*

Next we set this equal to zero to obtain:

*h = DÖ
(v*_{1}/*v _{o}) *

which is the answer.