# Two objects of equal mass, collision with friction

If two objects with equal mass collide on a floor that has friction, why is momentum not conserved?

I understand that in collisions where one object is initially at rest, the force of friction on both objects will be opposite. However, when two objects collide moving towards each other, the net force acting on both objects would be the same, so the change in momentum would be the same and thus momentum would be conserved. What am I missing here?

• Why do you say that momentum is not conserved? Mar 22, 2022 at 21:27

Momentum is always conserved when the net force on the whole system is zero. If the frictions are both the same at all times, but opposite in directions, then the net force is zero and momentum is conserved. However, i see where your confusion lies, the masses are losing velocity, at one point, say both have 5m/s and then after some time they will have 2 m/s, by $$p=mv$$ the momentum is not being conserved, right? Wrong, because momentum is a vector quantity, not a scalar.

Consider two masses going towards each other with velocities of equal magnitude. However, since velocity is a vector, and they are going towards each other, one of the velocities is going to be positive and the other, negative. The total momentum is, then: $$p_{tot} = mv +m(-v) = 0$$ $$\frac{dp_{tot}}{dt}= F^{NET} = d0/dt = 0$$

So, the calculations based on momentum are perfectly consisten with two friction forces of equal magnitude and opposite directions. The total momentum of the system in this case will be zero at all times, even if the particles stop, the momentum is still conserved, since it goes from 0 to 0.

But what if the velocities are different? Then the total momentum will be: $$p_{tot} = m(v_1) +m(v_2)$$ $$F^{NET} = 0 = m\frac{dv_1}{dt} + m\frac{dv_2}{dt}$$ $$\frac{dv_1}{dt} = -\frac{dv_2}{dt}$$

So, the acceleration of 2 is the negative acceleration of 1, again, this is consistent, because, since the forces are opposite, and the masses are constant, the accelerations must be opposite, but then you might say: But what if both particles stop again? The momentum will go from a non-zero value to zero, thus it's not conserved.

You have to remember that if the velocities are different, then after enough time, the particle with smaller magnitude of velocity will stop first, right? But at the moment it stops, the friction force at that particle goes to zero, and the only remaining force is the one that is being applied at the particle still moving, violating the net force condition, so, for different module of velocities, the conservation of momentum is only valid if both particles are still moving.

I think this clears all doubts you may have.

The momentum is conserved , if you consider the moment directly bevor and after the collision, after that the velocity will change because of friction