Finding the Differential Equation for a Free Falling Body with Air Resistance

[trojansc82]

Homework Statement

Under some circumstances a falling body B of mass m encounters air resistance proportional to its instantaneous 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).

 

[Doc Al]

What forces act on the object? What's the net force?

 

[trojansc82]

What forces act on the object? What's the net force?

Net force = mg + km?

 

[Doc Al]

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?)

 

[trojansc82]

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]

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.

 

[trojansc82]

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]

Net force = mg -kv ?

Exactly! Continue.