# How is momentum conserved in this example? [duplicate]

Suppose a sticky substance is thrown at wall. The initial momentum of the wall and substance system is only due to velocity of the substance but the final momentum is 0. Why is momentum not conserved?

• The final momentum is not $0$! – tfb Oct 10 '18 at 10:05
• What about the momentum changes of the system as you throw the substance? Is momentum conserved? Answer: momentum is always conserved, because conservation involved currents into and out of a system. Impulses change the momenta of systems, transferring momentum from one to another. Conservation does not equal constancy. – Bill N Oct 10 '18 at 15:50

You should also consider what the wall is attached to. And obviously it is the Earth. If we assume the Earth's velocity is zero after the substance is thrown, since there is the force that slow down the substance at the moment of impact, there is also the reaction force on Earth with the same magnitude and opposite direction. So Earth will gain velocity and final momentum of combined Earth and substance system will be equal to the intial momentum of the substance.

And also we can look at the situation in a bit different way. When we stand on the floor and throw the substance, there appears a friction force between our feet and the floor and it acts on us in the throw direction. So the friction force on Earth will be opposite to the throw direction and Earth will pick up speed towards the substance, too. And at any moment, Earth plus substance system will have zero momentum. The substance and the Earth will move towards each other and after the impact their speed will be zero.

If you assume that you throw the sticky substance from rest at the wall then your assertion that the total final momentum of the Earth/wall/you and sticky substance system is correct. Indeed that is also the initial momentum of the Earth/wall/you and sticky substance system before you threw the sticky substance.

In the act of throwing the sticky substance, the Earth/wall/you impart momentum on the sticky substance $$\vec p_{\rm ss}$$, and as a consequence of Newton's third law, the sticky substance exerts an equal magnitude opposite direction momentum on the Earth/wall/you $$\vec p_{\rm Ewy}$$ such that the initial momentum of the system $$0$$ is equal to the final momentum of the system, i.e. $$0 = \vec p_{\rm ss}+\vec p_{\rm Ewy}\Rightarrow \vec p_{\rm ss}=-\vec p_{\rm Ewy}$$

Assuming no air resistance, etc. the reverse happens when the sticky substance hits and sticks to the wall with $$\vec p_{\rm ss}+\vec p_{\rm Ewy}=0.$$

Of course you do not notice the movement of the Earth, etc. because it is so much more massive than the mass of the sticky substance.

In terms of magnitudes: $$m_{\rm ss} V_{\rm ss}= M_{\rm Ewy} v_{\rm Ewy} \Rightarrow v_{\rm Ewy} = \frac {m_{\rm ss}}{M_{\rm Ewy}}\times V_{\rm ss}\text{ and }\frac {m_{\rm ss}}{M_{\rm Ewy}}\ll1.$$

If you just consider the sticky substance already in motion, and the Earth/wall/you not moving before the sticky substance hits the wall, you have in terms of magnitudes:

$$m_{\rm ss} V_{\rm ss}= M_{\rm Ewyss} v_{\rm Ewyss} \Rightarrow v_{\rm Ewyss} = \frac {m_{\rm ss}}{m_{\rm ss}+ M_{\rm Ewy}}\times V_{\rm ss}\text{ and }\frac {m_{\rm ss}}{m_{\rm ss}+ M_{\rm Ewy}}\ll 1$$

with there being no noticeable movement after the collision.

Mind you, would you notice if the wall, still intact and connected to the Earth, did move given that you would also be moving whilst standing on the Earth?

The wall will move a little bit as well as exert a small force on whatever it's attached to, etc., etc., until you get to applying a force to the Earth. Everything else is so massive, so we can't see this happening. You are assuming an immovable wall, which is not physically the case.

Remember Newton’s 3rd law. The change in momentum is $$F \: \Delta t$$ (also known as impulse). So, since by Newton’s 3rd law the forces are equal and opposite then the change in momentum must also be equal and opposite.

Therefore, Newton’s laws guarantee conservation of momentum, and to see where the momentum goes all you have to do is look for the Newton’s 3rd law pair. So here momentum is transferred between the sticky substance and the wall, and the wall (being so massive) gains a little momentum which makes it move imperceptibly.