Find the equation for velocity as a function of time

[ggb123]

Homework Statement

A rock with mass "m" slides with initial velocity on a horizontal surface. A retarding force F that the surface exerts on the rock is proportional to the square root of the instantaneous velocity of the rock: F = -k * v^(0.5)

Find the equation for velocity as a function of time.

Homework Equations

F = ma
a = dv/dt

The Attempt at a Solution

I'm using online software and used up all my attempts at the answer, so it gave it to me:
vi = initial velocity
v = vi - ((vi)^0.5 * k * t)/m + (k^2 * t^2)/(4m^2)

And I have no idea how to get this. I'm thinking this is because I don't know much about integration. I'm hoping somebody could help me out in explaining this to me, even just a brief outline of the steps to the final answer would be great.

Any help is really appreciated, thanks

 

[rl.bhat]

$$F = -kv^{1/2}$$

$$a = \frac{dv}{dt} = -\frac{k}{m}v^{1/2}$$

$$\frac{dv}{v^{1/2}} = -\frac{k}{m}dt$$

Taking integration you get

$$2v^{1/2} = -\frac{k}{m}t + C$$

When t = 0, $$C = 2v_o^{1/2}$$

$$2v^{1/2} = -\frac{k}{m}t + 2v_o^{1/2}$$

Square both the sides and simplify.