Question

A soldier is tasked with measuring the muzzle velocity of a new rifle. Knowing the principles of projectile motion, he decides to perform a simple experiment at the indoor firing range. The soldier hangs a target a distance of d = 101 m from the end of the barrel. The rifle is mounted so that the bullet exits moving horizontally at the same height as the bullseye. After 6 trials, the soldier tabulates the values he measured for the distance, h, from the bullseye to the bullet strike.

Answer 1

The muzzle velocity is related to the distance to the target, D, and the time of flight t by

v = D /t

The bullet drop in that same time is

h = 1/2 g t^2, i.e.
t = sqrt(2h/g)

Therefore muzzle v and drop h are related as

v = D sqrt(g/(2h))

The first order derivative is

dv/dh= -1/2 *D sqrt(g/(2h^3)) = - v/(2h)

So the linear approximation of the error is

Δv = -v/(2h) * Δh

with h the average drop and Δh the uncertainty in h (measured as the standard deviation in h).