Question

A massive ball moving with speed $$v$$ collides with a tiny ball having a mass very much smaller than the mass of the first ball. If the collision is elastic, then immediately after the impact, the second ball will move with a speed approximately equal to:

A
v
B
2v
C
v2
D
infinite

Solution

The correct option is B $$2v$$By momentum conversation.$$MV+0={ MV }_{ 1 }+m{ V }_{ 2 }$$$$M\left( V-{ V }_{ 1 } \right) =m{ V }_{ 2 }\quad \longrightarrow (1)$$By energy conservation.$$\dfrac { 1 }{ 2 } M{ V }^{ 2 }+0=\dfrac { 1 }{ 2 } M{ V }_{ 1 }^{ 2 }+\dfrac { 1 }{ 2 } m{ V }_{ 2 }^{ 2 }$$$$\Rightarrow \quad M{ V }^{ 2 }=M{ V }_{ 1 }^{ 2 }+m{ V }_{ 2 }^{ 2 }$$$$M\left( { V }^{ 2 }-{ V }_{ 1 }^{ 2 } \right) =m{ V }_{ 2 }^{ 2 }\quad \longrightarrow (II)$$Dividing (II) by (I)$$\dfrac { { V }^{ 2 }-{ V }_{ 1 }^{ 2 } }{ V-{ V }_{ 1 } } =+{ V }_{ 2 }$$$$V+{ V }_{ 1 }={ V }_{ 2 }$$  putting in (II)$${ V }_{ 1 }=\left( { M }_{ 1 }-{ m }_{ 2 } \right) V/M+{ m }_{ 1 }$$$${ V }_{ 2 }=2MV/\left( M+m \right) \quad \quad \because \quad M>>m$$                                                  $$\therefore \quad M+m\approx M$$$$\boxed { { V }_{ 2 }=2V }$$Physics

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