Vertical Trajectory Calculation for Quadratic Drag of form -cv²

Air drag will depend upon the area of the object in motion and the resulting trajectory will depend upon its mass. For that reason the sphere calculation is typically the starting point for modeling motion with quadratic drag. Departures from spherical geometry can then be tried by varying the drag coefficient.

But a trajectory can be modeled with just the general form -cv² for the quadratic drag. This is an example of a vertical trajectory calculation.

Assume the object has a drag coefficient c = Ns²/m² and a mass of m = gm.

Under these conditions, it's terminal velocity will be
v_t = m/s
v_t = km/hr
v_t = mi/hr

and its characteristic time is
τ = s.

If this sphere is launched vertically with a velocity of
v₀ = m/s = ft/s

then on the way up at height y₁ = m = ft
it will have velocity v₁ = m/s = ft/s

It will reach a peak height y_peak = m = ft
at time t_peak = s

On the way down at height y₂ = m = ft
it will have velocity v₂ = m/s = ft/s

It will reach the ground at velocity
v_impact = m/s = ft/s = km/hr = mi/hr
at time t_impact = s.

For comparison, if there were no air friction the projectile would have reached height
h = m = ft
and would impact with the original launch velocity at time t = s.