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Air Gravity Free Fall

Let us consider the free fall of a large body, e.g. a parachute-diver in air, from a certain height h above earth. In case of bodies of large dimensions, the dominant force of fluid (air) friction is drag force which is proportional to square of the velocity:

F_fr = Av²

The drag force increases with velocity and at the instant, it is equal to force due to gravity, mg, the body attains a constant, limiting speed v_m given by,

Let us study the motion before the body achieves the limiting speed. The equation of motion is,

where we have taken all vectors positive vertically downwards. Integrating above equation, we find

The constant of integration C₁ is fixed by initial condition: at t = 0, v = 0, so that C₁ = 0. Hence, we get

That is, initially drag force is insignificant and body falls freely under gravity. With time, as velocity increases, the drag force becomes the decisive factor. For we put,

So that, v = v_m

The constant, is called the time constant of motion.

Experimentally, it is found that (for parachute-divers) vm is about 50 m/s in air. Hence, τ = 2.5 s. Empirically, therefore, we have

where t is in seconds. Solving numerically, we find that in about t = 10 s, the parachute-divers attain the limiting velocity of about 50 m/s.