Inelastic Sphere Collision!

Let after collision sphere moves with velocity in + ve X- direction and in + ve Y-direction translational and rotate with angular velocity $\omega$ in clockwise direction.

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Let Impulse due to normal is and friction are ${ J }_{ N }\quad \& \quad { J }_{ F }$ respectively . Then :

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${ J }_{ F }\quad =\quad \mu { J }_{ N }\quad \quad .\quad .\quad .\quad .\quad (1)$ .

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X-axis :

${ J }_{ F }\quad =\quad m{ v }_{ x }\quad -\quad 0\quad \quad \quad \quad .\quad .\quad .\quad (2)$ .

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Y-axis:

${ J }_{ N }\quad =\quad m{ v }_{ y }\quad -\quad m(-{ v }_{ o })\quad \quad .\quad .\quad .\quad .\quad (3)$ .

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Now , Using Angular momentum conservation about point of contact ( bottom most point )

$\cfrac { 2 }{ 5 } m{ R }^{ 2 }{ \omega }_{ o }\quad \quad =\quad \cfrac { 2 }{ 5 } m{ R }^{ 2 }{ \omega }\quad +\quad m{ v }_{ x }R\quad \quad \quad .\quad .\quad .\quad .(4)$ .

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Note : One can also use angular impulse equation instead of using Cons. of Angular Momentum

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Now using Newtons Law of experiment ( e=1/2 )

$e\quad =\quad \cfrac { -{ v }_{ o }\quad -\quad 0 }{ \quad \quad 0\quad -\quad { v }_{ y }\quad } \quad =\quad \cfrac { 1 }{ 2 } \\ \\ { v }_{ y }\quad \quad =\quad \cfrac { { v }_{ o } }{ 2 } \quad \quad .\quad .\quad .\quad .\quad .\quad (5)$ .

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Using all equations we get :

$\boxed { { v }_{ x }\quad =\quad \cfrac { 3 }{ 2 } \times \mu v_{ 0 }\\ \\ { v }_{ y }\quad \quad =\quad \cfrac { { v }_{ o } }{ 2 } \quad \\ \\ { \omega }\quad =\quad \cfrac { 4-15\mu }{ 4R } \times v_{ 0 } }$ .

Q.E.D

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Note : Here in this question we can't use Linear momentum conservation and energy conservation.

lets take the normal to N and time of contact be t ,as t is very-very small we can neglect gravity effect.

 let the initial velocity be q.

velocity along y axis =Y

velocity along x axis=X

• ue=Y

• Nt=m(Y-(-q))

• Nt(coeff. of friction)={2/5mR^(2)}(W-(-q/R))

• Nt(coeff. of friction)=m(X-0)

SOLVING ABOVE WE GET THE ANSWERS