Time to reach terminal velocity

[JC1]

Homework Statement

What is a formula I can use to calculate time to reach terminal velocity? Must account for air resistance during the time it took to reach terminal velocity.

Known:

mass of object
density of object
radius of object

magnitude of terminal velocity

height of fall (yes, it's high enough to reach tv)

variables necessary to calculate force of drag (done)

Homework Equations

I have no idea what equation I can use to calculate the time it would take to reach terminal velocity

The Attempt at a Solution

I have no idea how to attempt a solution without a formula to use. My text is awful.

 

[edgepflow]

You can apply Newton's 2nd law:

Fnet = ma = m dv/dt

Fnet = is the difference between the force of gravity and the drag force.

See if you can find equations for these.

Then you can sove

dv/dt = Fnet / m

for v(t).

 

[ideasrule]

If the drag force is proportional to v or v^2, which are the two most common scenarios, terminal velocity is never reached. Instead, the speed asymptotically approaches terminal velocity, and at some point becomes so close that, for all practical intents and purposes, it's equal to terminal velocity.

You can see this from the speed vs. time relationship for a F=-kv drag force. Integrating Newton's second law gives you an exponential relationship that asymptotically approaches mg/k, but never quite reaches it.