# Show that for a perfectly elastic collision co- efficient restitution is 1 . prove that.

1
by anielanism

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by anielanism

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Let two bodies of masses M and m travel with velocities U and u
before collision. Let them collide elastically. so energy is
conserved. Let the potential energy remain same for each of them. Let
them travel with V and v after collision.

Conservation of linear momentum:

M U + m u = M V + m v

M (V - U) = m (u - v) --- (1)

change in the momentum of one object is negative of the change in momentum of the other object.

Conservation of energy :

1/2 M U² + 1/2 m u² = 1/2 M V² + 1/2 m v²

M(V² - U²) = m (u² - v²) --- (2)

divide (2) by (1) to get:

V + U = u + v

V - v = - (U - u)

=> Relative velocity after collision = relative velocity before collision.

e = coefficient of restitution

= relative velocity after collision / relative velocity before collision

= 1

Conservation of linear momentum:

M U + m u = M V + m v

M (V - U) = m (u - v) --- (1)

change in the momentum of one object is negative of the change in momentum of the other object.

Conservation of energy :

1/2 M U² + 1/2 m u² = 1/2 M V² + 1/2 m v²

M(V² - U²) = m (u² - v²) --- (2)

divide (2) by (1) to get:

V + U = u + v

V - v = - (U - u)

=> Relative velocity after collision = relative velocity before collision.

e = coefficient of restitution

= relative velocity after collision / relative velocity before collision

= 1