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 v1 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 .
va = votana = voh/D
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/ v1 + L/ va
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/ v1 + D/( voh/D)
= h/ v1 + D2/voh
Now we differentiate:
dt/dh = 1/ v1 - D2/voh2
Next we set this equal to zero to obtain:
h = DÖ (v1/vo)
which is the answer.