
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
A uniform pool ball of radius  $r$  and mass  $m$  begins at rest on a pool table. The ball is given a horizontal impulse  $J$  of fixed magnitude at a distance  $\beta r$  above its center, where  $-1 \le \beta \le 1$ . The coefficient of kinetic friction between the ball and the pool table is  $\mu$ . You may assume the ball and the table are perfectly rigid. Ignore effects due to deformation. (The moment of inertia about the center of mass of a solid sphere of mass  $m$  and radius  $r$  is  $I_{cm} = \frac{2}{5}mr^2$ .)

[asy]
size(250);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
filldraw(circle((0,0),1),gray(.8));
draw((-3,-1)--(3,-1));
draw((-2.4,0.1)--(-2.4,0.6),EndArrow);
draw((-2.5,0)--(2.5,0),dashed);
draw((-2.75,0.7)--(-0.8,0.7),EndArrow);
label(" $J$ ",(-2.8,0.7),W);
label(" $\beta r$ ",(-2.3,0.35),E);
draw((0,-1.5)--(0,1.5),dashed);
draw((1.7,-0.1)--(1.7,-0.9),BeginArrow,EndArrow);
label(" $r$ ",(1.75,-0.5),E);
[/asy]

(a) Find an expression for the final speed of the ball as a function of  $J$ ,  $m$ , and  $\beta$ .

(b) For what value of  $\beta$  does the ball immediately begin to roll without slipping, regardless of the value of  $\mu$ ?

Present the answer in LaTex format: \boxed{Your answer}
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