In contrast to Exercise 45, the size of a falling lightweight object may be more significant than its mass when taking into account air resistance. One model that takes such an approach for falling raindrops is

where s(t) is the distance a raindrop has fallen (in meters), D is the raindrop diameter, and g = 9.8m/s2. Terminal velocity vterm is defined as the velocity at which the drop has zero acceleration (one can show that velocity approaches vterm as time proceeds).
(a) Show that v_term = √2000gD.
(b) Find vterm for drops of diameter 10⁻³ and 10⁻⁴ m.
(c) In this model, do raindrops accelerate more rapidly at higher or lower velocities?

Data From Exercise 45

According to one model that takes into account air resistance, the acceleration a(t) (in m/s²) of a skydiver of mass m in free-fall satisfies

where v(t) is velocity (negative since the object is falling) and k is a constant. Suppose that m = 75 kg and k = 0.24 kg/m.

Transcribed Image Text:

dfs dt² = 8- g 0.0005 D / ds dt 2