Two spheres are moving towards each other. Both have same radius but their masses are 2kg and 4kg. If the velocities are `4m//s` and `2m//s` respectively and coefficient of restitution is `e=1//3`, Find final velocities along line of impact.

Explanation

Let `v_(1)` and `v_(2)` be the final velocities of `A` and `B` respectively then by conservation of momentum along line of impact.

`m_(1)u_(1)+m_(2)u_(2)=m_(1)v_(1)+m_(2)v_(2)`
`2(4cos30^(@))-4(2cos30^(@))=2(v_(1))+4(v_(2))`
or `0=v_(1)+2v_(2)`....(1)
By Newton's Experimental Law.
`2(2cos30^(@))-4(2cos30^(@))=2(v_(1))+4(v_(2))`
or `0=v_(1)+2v_(2)`....(1)
By Newton's Experimental Law,
`e=("velocity of separation along LOC")/("velocity of approachalong LOC")`
or `(1)/(3)=(v_(2)-v_(1))/(4cos30^(@)+2cos30^(@))`
or `v_(2)-v_(1)=sqrt3` ...(2)
From the above two eqations,
`v_(1)=-(2)/(sqrt3)m//s`
Negative answer denotes that we have chosen the wrong direction, actual direction of final velocity will be opposite to the direction we assumed in figure
`v_(2)=(1)/(sqrt3)m//s`