Classical Mechanics - Drag Force

[zeromaxxx]

Homework Statement

Suppose a block sliding on a slippery horizontal surface experiences a drag force F=-cv^3/2 where c is a positive constant. At time t=0, the block is at position x=0 with initial positive velocity. Find the velocity and position as a function of time. Derive an expression for the limiting distance the block travels, or show that there is no limit.

Homework Equations

F=-cv^3/2

The Attempt at a Solution

For velocity:
I took the integral of the drag force

mdv = -cv^3/2 dt

int [dv/v^3/2]= -c/m * int[dt]

-2/v^1/2 + 2/v₀^1/2 = -c/m * t

Isolating for v:

v= [v₀^1/2/ 1+ (cv₀^1/2t/2m)]²

Can someone kindly check the arithmetic if I got it right.

As for position, I have trouble dealing with the integral of the velocity function. If someone can help me regarding this, it would be greatly appreciated.

 

[Sourabh N]

Int [dv/v^2/3] is not v^1/2 but v^1/3. See the error?

 

[zeromaxxx]

Int [dv/v^2/3] is not v^1/2 but v^1/3. See the error?

Yes, sorry there's a typo in the original question I put up, the drag force is F = -cv^3/2 and not 2/3. Sorry for this, I hope you can go through it again. Thanks!

 

[Sourabh N]

Okay. After you put v = dx/dt, the integral looks like Int [dx] = Int [dt/(a + b*t)²] where a and b are some trivial constants, right? That t integral is easy to do, you know how?

I am sleepy (read sloppy) right now, so I'd rather not fix those constants for you.

 

[zeromaxxx]

Hmm, it made more sense the way you put it though I still cannot determine the integral of t.

 

[Sourabh N]

A substitution a + b*t = w (some random variable) brings it in the form Int [dw/w²] with some constants I have suppressed. Does it look familiar now?