Vertical Trajectory

Objects moving at high speeds through air encounter air drag proportional to the square of the velocity. Describing the motion of objects under this quadratic drag usually requires numerical techniques rather than straight analytic formuli since the drag force and the gravitational force are not acting along the same line. The case of the vertical trajectory can be treated analytically since the forces are colinear. It is common practice to express the velocity and time in terms of the terminal velocity v_t and a characteristic time t.

Two common approaches to the quadratic drag force are:

1. Express the drag in terms of a single drag coefficient c:

2. Assume that drag is proportional to cross-sectional area and express in terms of area and a shape-dependent drag coefficient C:

Vertical Trajectory Calculation

For a sphere of radius r = cm and density r = kg/m3 r = gm/cm3 the mass is m = gm. Assume the object has a drag coefficient C = (the default value is C = 0.5 for a sphere) and is falling through air with density r* = kg/m3 (the default value is 1.29 kg/m3). Under these conditions, it's terminal velocity will be vt = m/s vt = km/hr vt = mi/hr and its characteristic time is t = 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.