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I know how to calculate the velocity of the center of mass before a collision. If a sphere of mass $m_1$, $u_1$ collides with a sphere of mass $m_2$, $u_2$, then the velocity of the center of mass before the collision is:

$$u_{cm} = \frac{m_1u_1+m_2u_2}{m_1 + m_2}$$

As I understand it, for an elastic collision where energy is conserved, then the velocity of the center of mass should remain constant.

If the velocities of the spheres are $v_1, v_2$ after the collision, then the velocity of the center of mass after the collision can be calculated as:

$$v_{cm} = \frac{m_1v_1+m_2v_2}{m_1 + m_2}$$

But because the collision is elastic $v_{cm}$ should equal $u_{cm}$ [again as I understand it, if the collision is inelastic then $v_{cm}$ would be less than $u_{cm}$].

The reason I am asking for confirmation on my understanding of the two equations above is because I am playing around with the 2-D collision simulator at https://www.physicsclassroom.com/Physics-Interactives/Momentum-and-Collisions/Two-D-Collisions-Simulator/Interactive. For the specific case where $m_1 = 10, m_2 = 1, u_1 = 2.83, u_2 = 2.83$ and the collision angle is 90 degree, the simulator provides $v_1 = 2.37$ and $v_2 = 5.64$. Note: running the simulator multiple times with the exact same initial conditions does result in some slight variations in the calculated velocities after the collision (for example $v_1 = 2.38 $ and $v_2 = 5.59 $).

Using either set of values for $v_1, v_2$, if I calculate the kinetic energy after the collision it is equal to the total kinetic energy before the collision - so that checks out. $u_{cm}$, which is before the collision, is equal to 2.83. However, if I use my 2nd equation above to calculate $v_{cm}$, which is after the collision, it is equal to 2.67 (for either set of listed values of $v_1,v_2$). So, for these numbers it seems like the velocity of the center of mass did not remain constant, despite the fact that there was no loss in calculated kinetic energy.

I'm trying to determine where the disconnect is. Is there a problem with my 2nd equation, or is there perhaps a problem with the online calculator. I've checked the calculations numerous times, so I don't think I made any calculational errors.

Qmechanic
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rdemo
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1 Answers1

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Since you are simulating collisions in 2D, the velocity variables $u$ and $v$ need to be in vector form. That is, the center of mass velocity equations will work for the x- and y-components of the velocities of the particles separately. If you plug in the speeds of the particles, the equations won't work in general.

Also, the collisions don't have to be elastic for the equations to work. Conserving center of mass velocity is another way if saying that momentum is conserved.

Mark H
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