Problem with Friction dependent on velocity, mostly a math problem.

Simen

Hello, first time here for me, I hope I posted in the right subforum.
I have a task at hand, just started a new physics course, but sadly I am far from deft with mathematics, and my physics book is in a different language and uses different notation than I am used to. The part I am having trouble with is the following:

1. The problem statement, all variables and given/known data


We have a block, with the starting velocity 5m/s.
The friction from the block is given as a = -μ|v|g
Where μ = 0.1 and g = 9.8 m/s^2

Find v(t)


2. Relevant equations
v(t) = v(0)+at

a = dv/dt = (d/dt)(dx/dt) =(d²x)/dt²


3. The attempt at a solution

As it stands now, I I realize I should integrate a in order to obtain v(t), but I am horrible at math, and do not know where to start, for some reason I keep ending up with an expression without v at all, but that does not help things either. If someone could show me the first steps in this that would be great. I've written a python program to solve it numerically as well, though I can not check if it is working, because I am too inept to get my analytical solution.

Would something like this be a step in the right direction?
a = -0.98|v|
v(0) = 5 m/s
a(0) = 0

dv/dt = -0.98*v

dv/dt² = -0.98*v*dt

(dv/dt²)1/v = -0.98dt

And then integrate from here? Could someone show me an example? Doesn't need to have any of my numbers, so long as I can learn some of what I need from it.

If anyone could provide any help, that would be greatly appreciated!

tiny-tim

Hello Simen! Welcome to PF!
(you left out a µ)

Now you "separate the variables" …

dv/v = -0.98µdt …

carry on from there

Simen

Thanks for the welcome:)

I actually didn't forget the μ, since μ = 0.1 and it was to be multiplied with g = 9.8 I just multiplied them before I started integrating.

Anyways, does this look right?
dv/dt = -0.98v

dv/v = -0.98dt

Integrate:

ln(v)-ln(v₀)=-0.98(t-t₀)

Clean up a little:
v/v₀=e^-0.98(t-t₀)

v = v₀e^-0.98(t-t₀)

Then putting in t₀ = 0 and v₀ = 5 and getting
v(t) = 5e^-0.98t

Does that look about right?

tiny-tim

(nice formatting, btw! )

Excellent!

(btw, always best to check by mentally putting the solution back into the original differential equation. to see if it comes out right … which it does! )