I am trying to find an optimal staging configuration of 'N' batteries to maximize flight time of my multirotor, which can shed its consumed batteries. The problem statement is as follows.
I have N batteries of masses $m_1, m_2, ... m_N$ in some order of masses. To maximize the flight time, we need to maximize the following expression, $$\sum_{k=1}^{N} \frac{a_k}{\left(1+\sum_{i=1}^k a_i\right)^{3/2}}$$
where $a_k$'s are some permutation of $m_j$'s (no repetitions).
Through some brute force computations, I conjecture that the optimal order is such that $a_1\leq a_2\leq ... \leq a_N$
Could someone help me prove this?