The stages of a two stage rocket have initial masses and respectively and carry a payload of mass . Both stages have equal structure factors and equal relative exhaust speeds. If the rocket mass, , is fixed, show that the condition for maximal final speed is
.
Find the optimal ratio when .
According to multi-stage rocket’s flight equation (see “Viva Rocketry! Part 2“), the final speed of a two stage rocket is
Let , we have
and,
Differentiate with respect to gives
It follows that implies
.
That is, . i.e.,
It is the condition for an extreme value of . Specifically, the condition to attain a maximum (see Exercise-2)
When , solving
yields two pairs:
and
Only (3) is valid (see Exercise-1)
Hence
The entire process is captured in Fig. 2.
Fig. 2
Exercise-1 Given , prove:
Exercise-2 From (1), prove the extreme value attained under (2) is a maximum.