Question

Application of the first order differential equations, from "Differential Equations", by Isabel Carmona.

2. A body with a mass of 9.7 kg is released from a height of 300 m without initial speed. The body finds an air resistance proportional to its speed.
If the speed limit must be 95 m/sec...
Find
A) body speed at a time t
B) the position of the body at a time t

The correct answers are:

A) v = 95 (1-e^{(- t / 9.7)})
B) x = 95t + 921.5 (e^{(- t / 9.7)} -1)

Please help me solving this problem step by step. Thank you so much.

Answer 1

Let us write the equation of motion for the given mass. There are two forces acting on it, one downwards that is mg and other opposing it's motion i.e. air resistance whose magnitude is proportional to its speed. Thus, let the net force on the body be F,

F=mg-kv, where k is a proportionality constant