# Conservation of Momentum = Conservation of Energy?

• Islam Hassan
In summary: However, the kinetic energy is not finite, it is just very small. So at this level, the two conservation laws of energy and momentum still differ by a factor of V/2.
Islam Hassan
Energy and momentum only differ by a factor of V/2. So in essence, at a fundamental level, does this mean that the conservation of energy is the conservation of momentum in disguise?

IH

Momentum is a vector, while energy is a scalar. There are elastic collisions where both are conserved, while only momentum is conserved in inelastic collisions. In inelastic collisions, energy and momentum are both carried away by particles we don't explicitly track. However, momentum is carried away equally in all directions, so that what is lost in one direction is "added back" by the loss in the opposite direction.

However, momentum is carried away equally in all directions, so that what is lost in one direction is "added back" by the loss in the opposite direction.
Please explain what you mean by this.

It is also not true that "Energy and momentum only differ by a factor of V/2" - that's kinetic energy, not total energy. Kinetic energy is not necessarily conserved.

Bill_K said:
Please explain what you mean by this.

I meant that in an inelastic collision, energy is lost as heat, say as radiation. This removes energy, but does not remove net momentum, even though radiation carries momentum, because the heat loss is non-directional.

atyy said:
Momentum is a vector, while energy is a scalar. There are elastic collisions where both are conserved, while only momentum is conserved in inelastic collisions. In inelastic collisions, energy and momentum are both carried away by particles we don't explicitly track. However, momentum is carried away equally in all directions, so that what is lost in one direction is "added back" by the loss in the opposite direction.

What about at atomic/particle physics level where *everything* is kinetic: heat, sound, etc. At this level, do the two conservation laws of energy and momentum essentially repeat the same message?

IH

No they don't. While momentum conservation follows, by definition, from spatial translation invariance since momentum is the conserved quantity of this symmetry, energy conservation is due to temporal translation invariance since energy is the corresponding conserved quantity of this symmetry.

A system might not be space-translation invariant (e.g., a particle in an external potential) but time-translation invariant (if the potential is time independent). Then energy is conserved but momentum isn't.

Islam Hassan said:
What about at atomic/particle physics level where *everything* is kinetic: heat, sound, etc. At this level, do the two conservation laws of energy and momentum essentially repeat the same message?

IH

Not everything is kinetic energy. Fundamentally, momentum is always conserved, but only total energy, not kinetic energy is always conserved. But to see where your intuition holds partially, let's restrict ourselves to collisions that conserve momentum and kinetic energy. Let's say we don't observe all the outgoing particles, and we find that we are apparently missing some momentum. This does mean that we will also be apparently missing some kinetic energy. However, we cannot infer the apparent kinetic energy loss purely from the apparent momentum loss and conservation of momentum, without using conservation of kinetic energy. For example, if there are 3 missing particles, two of them may be identical particles moving in opposite directions, carrying away kinetic energy without removing net momentum, while the third carries away both kinetic energy and momentum.

Last edited:
vanhees71 has got the right answer.
For example, a classical object in a constant (external) gravitational field. In this case, the z component of momentum is not conserved, but the x,y components of momentum are conserved. And since the potential doesn't explicitly depend on time, energy is also conserved.

Islam Hassan said:
What about at atomic/particle physics level where *everything* is kinetic: heat, sound, etc. At this level, do the two conservation laws of energy and momentum essentially repeat the same message?

IH

At the atomic/particle physics level, there is still potential energy.

## 1. What is the conservation of momentum?

The conservation of momentum is a fundamental principle in physics that states that the total momentum of a closed system remains constant. This means that the total amount of motion in a system does not change unless an external force is applied.

## 2. How is momentum conserved?

Momentum is conserved because in a closed system, any change in momentum in one direction must be balanced by an equal and opposite change in momentum in another direction. This is known as the law of action and reaction.

## 3. What is the relationship between conservation of momentum and conservation of energy?

The conservation of momentum is closely related to the conservation of energy, as energy is essentially the ability to do work. In a closed system, the total energy, including both kinetic and potential energy, remains constant. This means that if there is a change in momentum, there must also be a corresponding change in energy.

## 4. How does the conservation of momentum apply to real-world situations?

The conservation of momentum applies to all physical systems, including everyday situations such as collisions between objects and the motion of planets in the solar system. It is a fundamental principle that helps us understand and predict the behavior of objects in motion.

## 5. Can the conservation of momentum ever be violated?

In theory, the conservation of momentum can only be violated in situations where an external force is applied to a closed system. However, in practical situations, it is possible for momentum to appear to be lost due to factors such as friction and air resistance. In these cases, energy is converted into other forms, such as heat or sound, but the total amount of momentum and energy still remains constant.

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