Given this equation of acceleration dependient of velocity, how can I find an expression of velocity in function of time?

Feb 2020
1
0
North America
I could use some help with this problem. Im having trouble finding how to change the dependence from velocity to time in a).
Given the acceleration as a function of speed a = −kv (v - v_c) (a) Find an expression for speed as a function of time. (b) How would it look a sketch of velocity as a function of time for initial velocities v_o greater than v_c and less than v_c?
 

topsquark

Forum Staff
Apr 2008
3,049
648
On the dance floor, baby!
Okay, there's probably a simpler way to do this, so please, someone feel free to chime in.
\(\displaystyle a = \dfrac{dv}{dt}\)

so
\(\displaystyle \dfrac{dv}{dt} = -kv^2 + (kv_c) v\)

\(\displaystyle \dfrac{dv}{dt} - (kv_c) v = -kv^2\)

This is a Bernoulli differential equation. See here for how to solve it.

-Dan