Integrating velocity equation to find position?

[Cryptologica]

Homework Statement

For t < 0, an object of mass m experiences no force and moves in the positive x direction with a constant speed v_i. Beginning at t = 0, when the object passes position x = 0, it experiences a net resistive force proportional to the square of its speed: F_net = −mkv², where k is a constant. The speed of the object after t = 0 is given by
v = v_i/(1 + kv_it).

(a) Find the position x of the object as a function of time. (Use the following as necessary: k, m, t, and vi.)

(b) Find the object's velocity as a function of position. (Use the following as necessary: k, m, t, vi, and x.)

Homework Equations

a = Δv/Δt
v = Δx/Δt

The Attempt at a Solution

I am suspecting that I need to integrate the given function to find the position function? I know that v = Δx/Δt, so we should have:

dx/dt = v_i/(1 + kv_it)
then we need to take the definite integral from x₀ to x_f or just 0 to x_f. My calculus is a bit rough, but isn't this some sort of a separable differential equation? So we need to make all the x's and t's all on one side? Right now that seems, well, impossible since we have no x's? So do I need to find something involving v and x, then relate/substitute so I only have x's and t's? Help please...

 

[NascentOxygen]

Yes, integrate dx/dt = v_i/(1 + kv_it)

k and v_i are constants, so this is of the basic form 1/(1 + a.t)
and you just integrate with respect to t.

 

[Cryptologica]

Ok, thanks!