The law of conservation of momentum and change in momentum

[huey910]

If an object with mass m kg moves towards a wall with velocity u m/s, collides elastically with the wall and finally moves with a velocity –u m/s, then according to the equation of conservation of momentum: mu=-mu where the final momentum of the wall is negligible. This equation does not seem to make sense because this would imply that 1=-1. However, the above scenario happens all the time: where have my reasoning gone wrong?

 

[Doc Al]

The final momentum of the wall (plus the Earth it's attached to) is not negligible. Since the effective mass is so huge, the final speed would be very small but the total momentum is not.

Since the momentum of the moving mass changes by an amount equal to -2mu, the wall+earth must gain an equal and opposite amount: +2mu. Most of the time, who cares? But when learning conservation of momentum you need to care.

 

[ZapperZ]

If an object with mass m kg moves towards a wall with velocity u m/s, collides elastically with the wall and finally moves with a velocity –u m/s, then according to the equation of conservation of momentum: mu=-mu where the final momentum of the wall is negligible. This equation does not seem to make sense because this would imply that 1=-1. However, the above scenario happens all the time: where have my reasoning gone wrong?

You are forgetting the wall, which is attached to the earth.

If you look at the mass alone, of course there isn't a conservation of momentum of the mass-alone system, because a force (impact with the wall) acted on the system. The system that has the conserved momentum is the mass+wall+earth system, not the mass alone. So in principle, the wall+earth also moved back a little bit. But since the mass of wall+earth is so huge when compared to the mass of your object, you don't see wall+earth system move back to conserve the total momentum.

Zz.

 

[huey910]

Thank you very much

 

[PJC01]

The law of conservation of momentum states that in a closed system, the total momentum before and after a collision remains constant. In the scenario described, the wall is not considered a part of the closed system and therefore its momentum is not taken into account.

When we apply the conservation of momentum equation, we are only considering the momentum of the object before and after the collision. In this case, the momentum of the object before the collision is mu and after the collision is -mu, as the direction of the velocity changes. This does not mean that the momentum of the object is equal to the momentum of the wall.

In an elastic collision, the total kinetic energy of the system is conserved, but the individual momenta of the objects involved can change. In this scenario, the object loses its initial momentum and the wall gains an equal and opposite momentum, resulting in a total conservation of momentum in the closed system.

Therefore, the equation mu=-mu makes sense in this scenario and is a result of the conservation of momentum principle. The key is to consider the entire closed system and not just the individual objects involved in the collision.