Question

a gun is fired with muzzle velocity v0 = 1000 m/s straight up along the z-axis. While moving through the air the Earth's gravity pulls on the bullet, which has mass m = 20 grams and the bullet's air resistance force can be approximated by the relationship Fresist = -cv^2 where c is a constant = 2*10^-5 kg/m and where v=v(t) is the bullet's instantaneous speed in flight. Determine the expressions for the bullet's speed as a function of height z above the firing position during both the up-leg of its travel and during the down leg

Answer 1

define positive up
Vi = + 1000

going up
F = - m g - c v^2 = m dv/dt

dv/dt = -g -(c/m) v^2

v = Vi - g t -(c/m)integral of v^2 dt

z = Vi t - (1/2) g t^2 - (c/m) double integral of v^2 dt

I do not think there is an analytical solution. You must resort to numerical integration.