Part 1 - General Formula for an Object Falling in Air

We rewrite our differential equation:

in a more convenient form :

For convenience, let c be defined as:

So:

We can rewrite our differential equation as:

This is the same as:

Separating the variables:

Integrating both sides:

To perform this integration, we could either:

1. Factor the denominator, use partial fractions and then integrate (it needs the logarithm form), or
2. Use a table of integrals (integral #13), (easier); or
3. Use a computer algebra system, like Scientific Notebook. (easiest)

We obtain:

"K" is the constant of integration.

At t = 0, v = 0 so K = 0. Therefore:

Now we solve for v.

Multiply both sides by -2 and divide by c:

Take e to both sides to remove the logarithm:

Multiply out and solve for v:

Now, as t → ∞, the value of the fraction approaches −1, since e^-2gt/c → 0, giving us the terminal velocity v = −c.

So

is the terminal velocity for the falling object (in the downward direction).

Part 2 - Skydiver

To find c,

we use the given mass and the coefficient of drag for the skydiver.

mass = m = 80 kg

coefficient of drag = k = 0.2

So

The units are ms^-1, so the terminal velocity is approximately 225 km/h (1 ms^-1 = 3.6 km/h).

The graph of the velocity against time shows that it takes around 15 seconds to reach the terminal velocity:

Note:

1. The graph shows that the terminal velocity is never actually reached - the skydiver's velocityjust gets closer and closer to that velocity.
2. Actually, the air resistance changes as the air gets more dense nearer the Earth's surface. We have assumed it remains constant for this problem.
3. The human sky diver can change k easily by either spreading their arms and legs, or diving down with arms and legs tightly together.