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Discuss one dimensional perfectly inelastic collision of two bodies. Obtain the expression for their velocities after collision. Discuss the special cases.

Solution

Consider two bodies A and B of masses m₁ and m₂ moving along the same straight line with velocities u₁ and u₂ respectively.
Let the two bodies undergo perfect inelastic collision.
In a perfect inelastic collision, the two bodies stick together, therefore the relative velocity of separation of two bodies after the collision is zero.
Let v be the velocity of combined masses after collision.
By conservation of momentum,

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...(1)

rightwards double arrow

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...(2)
Special cases:
Case (i):
Two bodies are of equal masses.
Putting

space space straight m subscript 1 equals straight m subscript 2 equals straight m left parenthesis let right parenthesis

in equation (2), we get

rightwards double arrow

straight v equals fraction numerator straight u subscript 1 plus straight u subscript 2 over denominator 2 end fraction

Case(ii): When

straight m subscript 2

is at rest.
Putting

straight u subscript 2 equals 0

in equation (2), we get

rightwards double arrow

straight v equals fraction numerator straight m subscript 1 straight u subscript 1 over denominator straight m subscript 1 plus straight m subscript 2 end fraction

, is the velocity after collision.

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