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.