You need to specify kinetic friction coefficient and the initial velocity.
it says to answer it in terms of those coefficients
like, it's asking for formulas...
Initial w, angular velocity, is zero.
now while sliding
torque = I alpha
mu m g r = (2/5) m r^2 alpha
alpha = (5/2) mu g / r
F = m a
mu m g = m a
a = mu g
w = alpha t
v = Vi - a t
when does w r equal v (no slip)
r alpha t = v = Vi - a t
(5/2) mu g t = Vi - mu g t
(7/2)mu g t = Vi
t = (2/7)Vi/(mu g)
that is the time sliding
work back up to get distance and final v
Check that with energy argument:
frictional work done = mu m g d
Initial Ke - frictional work = Final Ke
(1/2) m Vi^2 - mu m g d = (1/2) m v^2 +(1/2) (2/5) m r^2 (v^2/r^2)
Vi^2 - 2 mu g d = (7/5)v^2
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