Modeling with First Order Equations

Question

A ball with mass 015kg is thrown upward with initial velocity 20m/s from the roof of a building 30m high. There is air resistance of magnitude v^2/1325 directed opposite to the velocity , where the velocity is measured in m/s.

a. Find the max height above the ground that the ball reaches.

b. Find the time that the ball hits the ground.

I am unsure how to set this one up.

Answer 1

What have you tried? How does gravity change the velocity? What about the air resistance?

Answer 2

Let us call $x(t)$ the position of the ball at time $t$ (the movement is in one dimension, we assume that $x(t)$ increses when the ball goes up ).

Newton's second law says that $$\frac{d^2 x}{dt^2} m = \sum {\rm forces}.$$

In this case, we have two forces:

1. The gravity, always going down: $-mg$.
2. The viscosity, opposed to the velocity. $$-\frac{1}{1325} \frac{dx}{dt}$$.

Thus, the equation of motion is $$\frac{d^2 x}{dt^2} m = -\frac{1}{1325} \frac{dx}{dt} -mg$$