A billiard ball of mass [itex]M[/itex] and radius [itex]R[/itex] is hit by a cue as shown in the figure.
The blow can be thought as an impulse [itex]J[/itex] of given value, and let [itex]μ[/itex] be the coefficient of static friction.
Find the maximum angle for which the ball's initial velocity isn't null.
The Attempt at a Solution
It seems I have a big, serious doubt here.
From the definition of Impulse I know that
[itex]J=\Delta p[/itex] (1)
[itex]JRsin(θ)= I\Deltaω[/itex] , with I being the moment of inertia of the body.
Since the body is at rest before being hit we can just write [itex]p[/itex] and [itex]ω[/itex] to indicate the initial values of the rotational and translational velocities.
This is where I get stuck: I fail to comprehend how to impose the non-null initial velocity, and thus how to get the disequation which will give me the maximum value of the angle.
The problem should be solved once I know which formula to use.
I know the velocity of a given point P is [itex]V= v_r+v_t=v_g+ωr_P[/itex], (rotational and traslational components, [itex]v_g[/itex] stands for the velocity of the c.o.m.) equation (1) can lead me to the value of [itex]v_g[/itex], but what about the other term?
That said the problem is saying that [itex]V[/itex], hence its magnitude, isn't zero, but I'm having trouble putting this into pratice.
I know this is a vague answer but I'm completely lost and I can't find useful examples on my notes.
The problem seems very easy, yet I can't solve it and this is pretty depressing; what am I forgetting about?
Forgive me for the bad pic but it's the best reproduction I can do at the moment, if you don't understand something feel free to say it.
Thank you for your help.