Collision of Two Bodies with Equal Mass and Opposite Velocities

When two bodies of equal mass collide with each other, the total momentum before the collision is equal to the total momentum after the collision. In this scenario, we have two bodies of mass 'm' moving with the same constant velocity 'v' but in opposite directions.

Total Momentum Before Collision:
The momentum of an object is given by the product of its mass and velocity. Therefore, the momentum of the first body before the collision is given by:
Momentum1 = mass * velocity = m * v

Similarly, the momentum of the second body before the collision is given by:
Momentum2 = mass * velocity = m * (-v) = -m * v (as the second body is moving in the opposite direction)

The total momentum before the collision is the sum of the momenta of both bodies:
Total Momentum Before Collision = Momentum1 + Momentum2
= m * v + (-m * v)
= 0

Total Momentum After Collision:
According to the law of conservation of momentum, the total momentum of the system remains constant if no external forces act on it. Therefore, the total momentum after the collision should also be zero.

Let's assume the common velocity of both bodies after the collision is 'v_f'. The momentum of the first body after the collision is given by:
Momentum1_after = mass * velocity_after = m * v_f

Similarly, the momentum of the second body after the collision is given by:
Momentum2_after = mass * velocity_after = m * (-v_f) (as both bodies move in opposite directions)

The total momentum after the collision is the sum of the momenta of both bodies:
Total Momentum After Collision = Momentum1_after + Momentum2_after
= m * v_f + (-m * v_f)
= 0

Conclusion:
Since the total momentum before the collision is zero and the total momentum after the collision is also zero, it implies that the common velocity of both bodies after the collision is zero. Therefore, the bodies come to rest after the collision.

Summary:
- When two bodies of equal mass collide with each other, the total momentum before the collision is equal to the total momentum after the collision.
- In this scenario, where both bodies have the same mass 'm' and are moving with the same constant velocity 'v' but in opposite directions, the common velocity of both bodies after the collision is zero.
- According to the law of conservation of momentum, the total momentum of the system remains constant if no external forces act on it.