Q.456. Elastic head-on collision.

Question 456.
A .45kg ice puck, moving east with a speed of 3m/s has a head on elastic collision with a .9 kg puck initially at rest. What will be the speed and direction of each object after the collision?

Answer 456.
For elastic collision, both the linear momentum as well as kinetic energies before and after collision are conserved.

Let u = velocity of the 0.45 kg ice puck
              after the collision towards the east
and v = velocity of the 0.9 kg ice puck
               after the collision towards the east

By the law of conservation of linear momentum,
initial linear momenta of the pucks before the collision
= momenta after the collision
=> (0.45) * 3 + 0.9 * 0 = 0.45 u + 0.9 v
=> u + 2v = 3 ... ( 1 )

By the law of conservation of kinetic energy,
(1/2) * (0.45) * 3^2 + 0 = (1/2) * (0.45) u^2 + (1/2) * (0.9) v^2
=> u^2 + 2v^2 = 9 ... ( 2 )

Plugging u = 3 - 2v from ( 1 ) in ( 2 ),
=> (3 - 2v)^2 + 2v^2 = 9
=> 6v^2 - 12v = 0
=> v (v - 2) = 0
=> v = 0 or 2 m/s
v cannot be zero => v = 2 m/s
and u = 3 - 2v = 3 - 2v = - 1 m/s

Answer:
Velocity of 0.45 kg puck is 1 m/s towards the west and
velocity of the 0.9 kg puck is 2 m/s towards the east.

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