# Where does the law of conservation of momentum apply?

Take the scenario of a snowball hitting a tree and stopping. Initially, the snowball had momentum but now neither the snowball nor tree have momentum, so momentum is lost (thus the law of conservation of momentum is violated?). Or since the tree has such a large mass, is the velocity of the tree is so small that it's hardly noticeable?

If the explanation is the latter, this wouldn't hold for a fixed object of smaller mass. So in that case, how would the law of conservation of momentum hold?

• To be clear: the conservation of momentum ALWAYS holds, in an incredible manner. If you will find a single exception, you're gonna be a Nobel Laureant; at least. But it says that the overall momentum is conserved, so as a check, always include the whole universe and remove all objects that are clearly not affected. Jan 8, 2021 at 14:40
• Possible duplicates: physics.stackexchange.com/q/175095/2451 , physics.stackexchange.com/q/59597/2451 and links therein. Jan 9, 2021 at 11:05
• Just to stress it more, conservation of momentum is valid even in nuclear reactions - one of the smallest objects possible (great versatility). Jan 9, 2021 at 11:43

Momentum is conserved only if there is no net external force on the system.

Consider the snowball and the tree as the system. In your case, the earth provides an external force on the tree, so the momentum of the snowball/tree system is not conserved. If the tree is "suspended" (not attached to the ground) momentum would be conserved, but the final velocity of the tree would be very small and hardly noticeable due to its large mass.

If the system is taken to be the snowball, tree, and earth, momentum is conserved , but the final velocity of the tree and earth (assuming the tree stays attached to the earth) is infinitesimally small due to the very large mass of the earth.

• Does this mean momentum isn't always conserved? I figured that the law of conservation of momentum would always apply in any situation? Jan 7, 2021 at 17:43
• So the momentum is conserved in the earth is included as part of the system (e.g. the momentum of the snowball is transferred to earth?) Jan 7, 2021 at 17:45
• Yes, it depends on whether the earth is part of the system you are evaluating. Jan 7, 2021 at 17:47
• That makes sense. So if the earth is part of the system, the force of friction by the earth on the tree is no longer external, thus there is no net external force so momentum can be conserved Jan 7, 2021 at 17:49
• If the system consists of the snowball and the tree/earth, momentum is conserved. It's simply not possible to "measure" the momentum following the collision given that the velocity of the system following the collision is too small to measure. Jan 7, 2021 at 17:52

Momentum of the body will be transferred to earth, let's exaggerate the numbers in your favor and assume your snowball is $$10kg$$ and you managed to throw it with $$v_s = 200\frac{m}{s}$$ where subscript "s" stands for snowball. Mass of the earth is approximately $$5.97\cdot10^{24}kg$$. For simplicity assume it was an elastic collision

$$m_sv_s = m_ev_e$$

where "e" stands for the earth. If you plug the numbers in, thus you get

$$v_e \approx 3.3\cdot10^{-22}\frac{m}{s}$$

• Maybe good to mention that this is also the momentum loss of the earth when the snowball is released. So it's not like we can accelerate the earth by throwing a lot of snowballs. Jan 7, 2021 at 18:20
• I think completely inelastic collision might be more accurate for snowballs. Snowballs are not really know from their elasticity. Jan 9, 2021 at 5:38
• @Bernhard: What if everybody jumped? Jan 9, 2021 at 17:05

The tree is attached to the earth. The momentum from the snowball is transferred to the tree and then distributed troughout the whole earth. Because a tree is a solid object this transfer happens almost instantaneously. The earth is so large that this tiny amount of momentum won't be noticable at all.

Let give a bigger picture. Assume the snowball appears (due to atmospheric processes) at some height "h". From this moment, gravity accelerates it against earth (discard wind forces). In the same way, earth is accelerated towards the snowball. Gravitational forces are the same for snowball and earth in modulus but opposite sense, same for momentum, accelerations are not. Total momentum is always zero.

After some time, the snowball and earth collides. Their momentum cancel, final velocity after collision vanishes and the final object snowball+earth is again at rest (regarding this subject), returning to its initial stage.

The ground applies a frictional/constraint force , so there is a net external force on the system and hence, Conservation of momentum can't be used on the snowball and tree as a system.

The snowball will likely break apart and some of that momentum will be transferred to its subsequently far-flung constituent far flung parts. It will probably deform, transferring some of that momentum into the new shape. Also, some of that kinetic energy will be converted to heat, both into the tree and into the material of the snowball itself. Then there is the sound produced when the snowball hits, taking some of the energy of momentum away with it.

• Please note that energy and momentum are both independently conserved. Getting rid of the energy does nothing to get rid of the momentum. Jan 9, 2021 at 1:22

Momentum is always conserved, even when there are forces, if you take into account every object interacting through forces as part of you system. The only time it will seem to be violated is if you exclude part of the system which is interacting with the system that you are considering, even if its through a long distance force like gravity. In your example the missing part of the system would be the earth, as the earth is ultimately where the momentum transfers to. But because of the mass of the earth, the velocity that it gains is very very small. Furthermore the initial force that you applied to throw the ball had made the earth move in the opposite direction at the start. So starting from that point in time, the net momentum gained by the earth will be 0. But then the initial momentum would also be 0 and so will the final momentum.

If you want to go further into the concept, the law of conservation of momentum is a direct consequence of Newton’s second law, which is that the change in momentum of an object is equal to the force applied on it, and Newton’s 3rd law, which is that every action force has an equal and opposite reaction force. You should be able to see how that leads to the conservation of momentum. If you take into account the whole system starting from the time everything was at rest before each object gains its initial momentum via forces, the total momentum of the system will always be 0.