AP Physics differential equation

[MrPyromaniac]

Homework Statement

A block of mass m, which has an initial velocity v0 at time t = 0, slides on a horizontal surface. If the sliding friction force f exerted on the block by the surface is directly proportional to its velocity (that is, f = -kv),
A) Write a differential equation for the velocity v in terms of the given variables and fundamental constants.
B) Find the acceleration a of the block in terms of m, k, and v.
C) The speed v of the block as a function of time t.
D) The total distance the block slides.

Homework Equations

F = m*a

The Attempt at a Solution

I understand why B is a = (-kv)/m, and I know the answer to a is dv/dt = -kv/m but I don't know why, and I have no idea how to solve for velocity. Assuming I get a function for the velocity, what general equation would I use to solve for the total distance? I originally thought x-x0 = v0*t + (1/2)*a*t^2, but the acceleration isn't uniform.

 

[axmls]

You have two options:
1. You can use separation of variables if you're familiar with it.
2. The solution to linear ODEs with constant coefficients is found by assuming v=e^{s t} and finding out what s is.

 

[haruspex]

I understand why B is a = (-kv)/m, and I know the answer to a is dv/dt = -kv/m but I don't know why,

What equation connects a with v?

and I have no idea how to solve for velocity.

Rearrange the equation to have all the references to v on one side and all references to t on the other. This is the "separation of variables" method axmls refers to.

Assuming I get a function for the velocity, what general equation would I use to solve for the total distance?

You did well to realize you cannot use the SUVAT equations here.
Suppose you have velocity as a function of time, v=f(t). What differential equation relates velocity to distance? (This is very like the first question I asked you in this post.)