A classical mechanics problem by Arpita Karkera

When the A will be at the highest position, its vertical velocity will be zero and horizontal velocity will be equal to that of B. (Let mass of each body be m)

Linear momentum will be conserved in horizontal direction,

$mv=m{ v }_{ 1 }+m{ v }_{ 1 }$

${ v }_{ 1 }=\frac { v }{ 2 }$

Total mechanical energy is conserved,

$\frac { 1 }{ 2 } I{ \omega }^{ 2 }+\frac { 1 }{ 2 } m{ v }^{ 2 }+0=\frac { 1 }{ 2 } I{ \omega }^{ 2 }+\frac { 1 }{ 2 } m{ { v }_{ 1 } }^{ 2 }+\frac { 1 }{ 2 } m{ { v }_{ 1 } }^{ 2 }+mgh$

${ v }^{ 2 }=2{ { v }_{ 1 } }^{ 2 }+2gh$

${ v }^{ 2 }=\frac { { v }^{ 2 } }{ 2 } +2gh$

${ v }^{ 2 }=4gh$

$h=\frac { { v }^{ 2 } }{ 4g }$

a=1 and b=4 a+b=5