# In the collision between two bodies a heavier and the other lighter. Write the relation between the changes in momentum of two bodies

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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 --- (3)*

** V - v = - (U - u) --- (4)**

Relative velocity after collision = relative velocity before collision.

For finding the change in momentum of the bodies: we have to find V and v in terms of u and U.

** V = v + u - U ** --- (5)

Substitute this in equation (1)....

M (v + u - 2 U) = m u - m v

* v *= (m u - M u + 2 M U) / (M + m)

* v =* **u + 2 M (U - u) / (M+m) --- (6)**

by (5) then** V = 2 u - 2 [ M u + m U ] / (M + m)**

V = [2 m u - m U + M U ] / (m + M) --- (7)

The change in momentum of each body:

M (V - U) = 2 M m (u - U) / (M + m)

m (v - u) = 2 m M (U - u) / (m + M)

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some notes.... if the masses are equal and then the initial velocities are equal and opposite...then: ie., M = m and U = -u then:

V - U = u - U => V = - U

v - u = U - u = 2 U => v = - u

Conservation of linear momentum:

M U + m u = M V + m v

Conservation of energy :

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

divide (2) by (1) to get:

Relative velocity after collision = relative velocity before collision.

For finding the change in momentum of the bodies: we have to find V and v in terms of u and U.

Substitute this in equation (1)....

M (v + u - 2 U) = m u - m v

by (5) then

V = [2 m u - m U + M U ] / (m + M) --- (7)

The change in momentum of each body:

m (v - u) = 2 m M (U - u) / (m + M)

============================

some notes.... if the masses are equal and then the initial velocities are equal and opposite...then: ie., M = m and U = -u then:

V - U = u - U => V = - U

v - u = U - u = 2 U => v = - u