Two objects slide over a frictionless horizontal surface The first object, mass `m_(1)=5` kg, is propelled with a speed `u=4.5m//s` towards the second object, mass `m_(2)=2.5` kg while is intially at rest. After the collision, both objects have velocities which are directed at `theta=30^(@)` on either side of the original line of motion of the first object. What are the final speed of the two object? Is the collision elastic or inelastic?

Explanation

Let us adopt the conrdinate system shown in diageram
Conservation of momentum along x-axis yields
`m_(1)u=m_(1)v_(1)costheta+m_(2)v_(2)cos theta" "…(1)`
Likewise, conservation of momentum along the y-axis yields
`0=-m_(1)v_(1)sin theta+m_(2)v_(2)sintheta`
`or m_(1)v_(1)sin theta=m_(2)v_(2)sin theta" "...(2)`
From equation (1) & (2), we obtain

`v_(1)=(u)/(2costheta)`
`=(4.5)/(2cos30)`
`=2.6m//s`
`and v_(2)=(m_(2)v_(1))/(m_(2))`
`=(5xx2.6)/(2.5)`
`=5.2m//s`
The initial kinetic energy of the system is
`k_(i)=1/2m_(1)u^(2)`
The final kinetic energy of the system is
`K_(f)=1/2m_(1)v_(1)^(2)+1/2v_(2)^(2)`
`=1/2m_(1)((u)/(2cos30))^(2)+1/2m_(2)((m_(1)V_(1))/(m_(2)))^(2)`
`=1/2m_(1)u^(2)[(1)/(4cos^(2)30)+(5)/(2.5xx4xx3/4)]`
`=1/2m_(1)u^(2)[(1)/(2)+(2)/(3)]=1/2m_(1)u^(2)`
Since `K=K_(f)`, the colision is elastic