# Free falling body question

## Homework Statement

Under some circumstances a falling body B of mass m encounters air resistance proportional to its instantanous velocity v. Use Newton's second law to find the differential equation for the velocity of v of the body at time t. Recall that acceleration a = dv/dt. Assume in this case that the positive direction is downward.

## Homework Equations

F = ma

a = dv/dt

air resistance = kv

kv = mg

## The Attempt at a Solution

kv = mg (dv/dt)

(dv/dt) (k/m) v = g

The solution in the text is (dv/dt) + (k/m) v = g.

I'm failing to understand why (dv/dt) is added to (k/m).

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Doc Al
Mentor
What forces act on the object? What's the net force?

What forces act on the object? What's the net force?
Net force = mg + km?

Doc Al
Mentor
Net force = mg + km?
mg, yes. But what is 'km'? There's a force proportional to the velocity, so show that. (What direction--and thus sign--does it have?)

mg, yes. But what is 'km'? There's a force proportional to the velocity, so show that. (What direction--and thus sign--does it have?)
So then v = mg?

I'm really confused....I don't feel like this question provides enough information to set up an equation.

I'm only able to set up this equation:

F = ma, letting F = kv, m = mg, and a = (dv/dt).

kv = mg (dv/dt)

Simplified:

(dv/dt) (k/m) v = g

Doc Al
Mentor
So then v = mg? The units don't even match. (You can't set a velocity equal to a force.)

I'm really confused....I don't feel like this question provides enough information to set up an equation.

I'm only able to set up this equation:

F = ma, letting F = kv, m = mg, and a = (dv/dt).
There are two forces acting: One is the weight (mg); the other is the air resistance, which you correctly have as kv (but what sign should it have?).

(Setting m = mg doesn't make much sense!)

So write the two forces, add them up to get the net force (pay attention to the signs of the forces), then apply ΣF = ma = m(dv/dt). That will give you the equation you need.

There are two forces acting: One is the weight (mg); the other is the air resistance, which you correctly have as kv (but what sign should it have?).
Net force = mg -kv ?

Doc Al
Mentor
Net force = mg -kv ?
Exactly! Continue.