I know the velocities (v) of an object after it fell for respective distances (s). From that alone, is it possible to calculate the drag force at a specific moment in time? If not what else would I need to find? So far I have looked at the linear air drag hypothesis and also the following equation, however the falling object I am dealing with is not a sphere. $$F_D=\frac{1}{2}C_D\rho_{air}Av^2$$
Thanks in advance.
Preliminary information on the problem can be found in every fluid dynamics textbook or on line. See for example wikipedia.
However, a key point to take into account when using the expression $$ F_D=\frac{1}{2}C_D \rho_{air} A v^2 $$ is that $C_D$ is constant only over limited interal of velocity. For example, in an answer to a related question you may see a plot of $C_D$ for a sphere, as a function of the Reynolds number, which is proportional to the flow velocity $v$.
So, while from the final velocity of a falling object it is possible to extract the value of $F_D$ and then of $C_D$, strictly speaking, that value of $C_D$ will be valid only for the corresponding Reynolds number. It is clear that at very low Reynolds numers ($Re \ll1$) a Stokes-like formula where drag force is proportional to $v$ should be more appropriate. In general, one has to measure $F_D$ at different velocities, in order to assess the value of $C_D$.
