Q.73. Collision Problem.

Question 73.
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 73.
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

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