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Bullet pendulum problem w/ Calculus - AP Physics C

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

See problem here:


2. Relevant equations

p = m * v
K = .5 * m * v^2
U = m * g * h
a = dv/dt
v = dx/dt

3. The attempt at a solution

I've gotten A and C fine, I'm still working to get B. D is the trouble part.

F_net = m * a
Only force acting is -bv

m * a = -bv
m * dv/dt = -bv

dv/dt = -bv / m

dv = -bv/m * dt

dv/v = -b/m * dt

At this point the published solution makes no sense to me at all. I am used to just putting an indefinite integral on both sides at this point. My confusion is about the bounds. Why is the left side v_0 to v?

I understand the steps after setting up the integrals (including evaluating the integrals and using log properties and so on).
Okay: I finally arrived at the right answer by just using indefinite integrals and including a constant of integration that I solved for. That got me v(t) = v_0 * e ^ (-bt/m)

Then I did the same thing to get position from velocity, and used indefinite integrals again...
I got position = m*v_0 / b * (1 - e^(-bt/m))

Taking the limit as t-> infinity (b/c the block never fully stops, v(t) has no zeros but does a horizontal asymptote) yields m*v_0/b, which is the answer.

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