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

A ball is released along a horizontal surface with a co-efficient of friction Us at a speed V. Find the time it takes for the ball to start rolling (leave the moment of inertia as I).

2. Relevant equations

Quite a few, all of them are stated in my working (that I believe are relevant).

After reviewing my lecture notes I found that I copied down a different solution. I attempted to redo the problem but came up with this solution, I must of either copied the answer incorrectly or made a mistake in the above working.

In regards to reasoning The Torque net must be equal to the frictional force acting directly below the ball multiplied by the radius, R. I assumed that the initial rotational velocity was zero. Using Torque(net) = I x alpha I found an expression for alpha. As the initial rotational velocity was equal to zero alpha = w/t.

I made the assumption that then the ball stopped rolling, V=Rw and solved for V. Using the kinematic equation v = u + at I solved for u (the initial linear velocity) and then solved for t.

Your method and result are both correct. Be careful with the signs, sometimes you wrote them incorrectly. The friction accelerates rotation but decelerates translation: Initially the ball slides, so α=dω/dt=μmg/I, ω=μmgt/I. μ is the coefficient of kinetic friction. At the same time, friction decelerates the translational motion of the CM: a=-μg, v=v_{0}-μgt.