# Momentum conservation vs center of mass [closed]

We have, if there is no net external force,momentum of the system is conserved,

the same thing we get from the motion of center of mass,that is the velocity of the center of mass remains constant,

so are these two different concepts or just another way to explain the same thing.

The velocity of the center of mass is simply the momentum divided by the mass. So they are just two different ways to say the same thing, simply scaled by mass.

• so,momentum is a physics concept where as center of mass is a mathematical concept to simplify the physics concept. Sep 2, 2022 at 19:57
• I didn't make that distinction and I wouldn't
– Dale
Sep 2, 2022 at 20:12
• that was just my assumption as i have come across discussions where it started that center ot mass is a mathematical concept. Sep 2, 2022 at 20:58

Suppose a system of N particles with a total mass $$M$$. The centre of mass, from a point in an inertial frame of reference is by definition: $$\mathbf r_{cm} = \frac{m_1\mathbf r_1 + m_2\mathbf r_2 +...+ m_n\mathbf r_n}{M}$$

If it moves at a constant velocity:$$\frac{m_1\mathbf r_1 + m_2\mathbf r_2 +...+ m_n\mathbf r_n}{M} = \mathbf r_0 + \mathbf v_{cm}t$$

Differentiating both sides: $$\frac{m_1\mathbf v_1 + m_2\mathbf v_2 +...+ m_n\mathbf v_n}{M} = \mathbf v_{cm}$$

The numerator of the left side is the momentum of the system, and as we supposed the velocity of the centre of mass constant, that quantity is also constant.

This is the problem with unmotivated definitions.

Consider an inertial frame K with respect to which n particles are moving each with some velocity $$\vec v_i$$ and mass $$m_i$$, the net momentum would be

$$\vec P_K=\sum m_i\vec v_i$$ Now we ask the question in which frame do we get the net momentum of these particles to be zero? Let this frame move with velocity $$\vec V$$ wrt K and we'll call this frame G, thus by definition $$\vec P_G=\sum m_i\vec v_i'=0$$ Where $$\vec v_i'$$ is the velocity of particle wrt to G.

We can make a relationship between velocities of particles in K and G as follows

$$\vec v_i=\vec V+\vec v_i'$$ Putting this in momentum equation of G we get $$\sum m_i(\vec v_i-\vec V)=\sum m_i\vec v_i-\sum m_i\vec V=\vec P_K-\sum m_i\vec V=0$$

Or $$\vec P_K=\sum m_i\vec V=M\vec V$$ Where $$\sum m_i=M$$ since V is a constant. This is equivalent to saying that $$\vec V= \frac{\sum m_i\vec v_i}{M}$$

Which on integration gives center of mass.

Thus indeed if $$P_K$$ is a constant then so is center of mass and they are equivalent ways of saying the same thing.

no net external force,

momentum of the system is conserved,

This should read "total momentum of the system is conserved," and maybe that is what you meant. The individual particle momenta can change due to internal forces.

The total momentum of a system of particles with masses $$m_i$$ is: $$\vec P = \sum_i m_i \vec v_i$$

the velocity of the center of mass remains constant

The center of mass is: $$\vec R = \frac{\sum_i m_i \vec r_i}{\sum_j m_j}$$

As long as the particle masses remain constant, the velocity of the center of mass is: $$\vec V = \frac{\sum_i m_i \vec v_i}{\sum_j m_j}$$

Clearly, in this case, the velocity of the center of mass is related to the total momentum by $$\vec V = \frac{\vec P}{M}\;,$$ where $$M$$ is the total mass $$M=\sum_j m_j$$.

So, if the particle masses are conserved, then conservation of total momentum is effectively the same as the velocity of center of mass remaining constant (because "remain constant" is what "conserved" means).

• yes,but I wanted to know why there are two concepts meaning the same thing. Sep 2, 2022 at 20:09
• They don't mean exactly the same thing. They are very close to the same thing, and are effectively the same thing under certain circumstances, which I explained in my answer.
– hft
Sep 2, 2022 at 20:53
• You might as well ask why we differentiate between Hesperus (Vesper) and Phosphorus (Lucifer), which are the same thing (they are both the planet Venus).
– hft
Sep 2, 2022 at 20:56