A shell of mass `2 kg` moving at a rate of `4 m//s` suddenly explodes into two equal fragments. The fragments go in directions inclined with the original line of motion with equal velocities, If the explosion imparts `48 J` of translational kinetic energy to the fragments, find the velocity and direction of each fragment.

Explanation

Conservation of linear momentum of along `X` and `Y` axes gives
`"mu"=m_(1)v_(1) costheta_(1)+m_(2)v_(2)costheta_(2)`
i.e. `2u=1xxv_(1)costheta_(1)+1xxv_(2)costheta_(2)`
`implies v_(1)costheta_(1)+v_(2)costheta_(2)=2u`…………i
and `0=m_(1)v_(2)sintheta+m_(2)v_(2)sintheta_(2)`
As `m_(1)=m_(2)=1kg`
`implies v_(1)sintheta_(1)=v_(2)sintheta_(2)`
As `v_(1)=v_(2)=v`(say)
`implies theta_(1)=theta_(2)=theta` (say)
from eqn i
`implies v costheta+vcostheta=2u`
`implies v costheta =u=4`...............ii

Also conservation of energy gives
`1/2mu^(2)+48=1/2(m_(1))v^(2)+1/2(m_(2))v^(2)`
`implies 1/2xx2xx(4)^(2)+48=1/2xx1v^(2)+1/2xx1v^(2)`
or `v^(2)=64`
`implies v=8m//s`
`costheta=u/v=4/8=1/2`
`implies theta=60^(@)`