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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
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B
2v
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C
v2
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D
infinite
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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)
$${ V }_{ 1 }=\left( { M }_{ 1 }-{ m }_{ 2 } \right) V/M+{ m }_{ 1 }$$
$${ V }_{ 2 }=2MV/\left( M+m \right) \quad \quad \because \quad M>>m$$
                                                  $$\therefore \quad M+m\approx M$$
$$\boxed { { V }_{ 2 }=2V } $$

951406_597963_ans_ad37267eaaac472f99eaebfeb6754dad.jpg

Physics

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