Collision Problem

Q. Collision Problem.

Question
A car moving at speed v undergoes a one-dimensional collision with an identical car initially at rest. The collision is neither elastic nor fully inelastic; 2/17 of the initial kinetic energy is lost. 
Find the velocities of the two cars after the collision. Express your answer in units of v.

Answer

Let equal mass of both cars = m
Initial velocity of 2nd car, u = 0
Let v' and u' be the final velocities of the 1st and 2nd car respectively.

Momentum is always conserved.
So, mv + 0 = mv' + mu'
=> v = v' + u' ... ( 1 )

2/17 of initial K.E. is lost => 15/17 of initial K.E. = Final K.E.
=> (15/17)(1/2)mv^2 = (1/2)mv'^2 + (1/2)mu'^2
=> (15/17) v^2 = v'^2 + u'^2 ... ( 2 )

From equations ( 1 ) and ( 2 ),
(v' + u')^2 - (v'^2 + u'^2) = v^2 - (15/17)v^2
=> 2v'u' = (2/17) v^2
=> (v' - u')^2 = (v' + u')^2 - 4v'u' = v^2 - (4/17)v^2 = (13/17)v^2
=> v' - u' = v √(13/17) = 0.874 v ... ( 3 )

Solving equations ( 1 ) and ( 3 ),
v' = 0.987 v and u' = 0.063v