JEE Piston's paradoxes

Two piston with masses and cross-section as shown . After an explosion in the space b/w the piston , the pistons fly out of pipes.Velocity of left piston is given in diagram. Then How fast is 2nd piston moving? Chose correct matching .

\(Case(i)\): If pipe is fixed and can not move .

\(\displaystyle{1)-\cfrac { { m }_{ 1 }v }{ { m }_{ 2 } } ,2)-\cfrac { { S }_{ 2 }v }{ { S }_{ 1 } } ,3)-\cfrac { { m }_{ 1 }{ S }_{ 1 }v }{ { m }_{ 2 }{ S }_{ 2 } } ,4)-\cfrac { { m }_{ 1 }{ S }_{ 2 }v }{ { m }_{ 2 }{ S }_{ 1 } } }\)

\(Case(ii)\) If pipe is not fixed and its total mass is M ?Neglect any friction between the piston and tube wall.

\(\displaystyle{1)-\cfrac { { m }_{ 1 }v }{ { m }_{ 2 }+M } ,2)-\cfrac { { S }_{ 2 }v }{ { S }_{ 1 } } ,3)-\cfrac { { m }_{ 1 }{ S }_{ 1 }v }{ { m }_{ 2 }{ S }_{ 2 }+M({ S }_{ 2 }-{ S }_{ 1 }) } ,4)-\cfrac { { m }_{ 1 }{ S }_{ 2 }v }{ { m }_{ 2 }{ S }_{ 1 }+M({ S }_{ 2 }-{ S }_{ 1 }) } }\)

Answer as the sum of two values of respective options.

For example, if you got answer as option 1 in case (i) and option 2 in case (ii) then your answer is 1+2 = 3.

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