# Derive a first order ODE equation for a falling body, and use it to design a suitable parachute for a given problem area.?

### 2 Answers

- Ian HLv 72 months ago
For larger spherical/streamlined objects, air resistance is roughly

proportional to velocity squared and so, taking downward velocity positive,

mdv/dt = mg - kv^2 (where k is the coefficient of drag)

dv/dt = g - (k/m)v^2 ......................................(1)

dv/dt = 0 immediately gives a terminal velocity F

F = √(mg/k) ....................................................(2)

If we take m = 80 kg, g = 9.81 and k = 0.2 we get

a terminal velocity of 62.6 m/sec ~ 225 km/hr ~ 140 mph

That is the result without a parachute. We need to greatly reduce the terminal velocity. We need the equivalent coefficient of drag, K say, of man and parachute to be much larger. As an example if K = 2500k we get a terminal velocity of about 4.5 km/hr which seems reasonable.

If this was a school project, the coefficient of drag of a parachute could be found by experiment. See this: -