# The law of conservation of momentum and change in momentum

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
Mentor
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
Staff Emeritus
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.

Thank you very much