ENERGY AND MOMENTUM: Elastic Collisions and the Center of Mass Velocity ( Part 1 ) 

A system’s center of mass is the location where the average mass distribution of the system is located. Consider a system in which two equal masses rest at the ends of a balance:

The sum of torques acting upon the system is zero, and it will remain in static equilibrium until acted upon by an outside force. For all practical purposes, the mass distribution of the system can be considered to exist at the system’s midpoint:

The center of mass location within a system is calculated as follows:

c.o.m. = ∑m_tx_t /M

First, a point of reference within the system is chosen. After a reference point has been chosen, the product of each mass ( m ) and its position ( x ) relative to the point of reference is taken. These values are added together prior to being divided by the total mass ( M ) of the system. In a system of two masses, the point of reference is the position of one of the masses. A two-mass system’s c.o.m. value is determined as shown below:

m₁x₁ = ( m₁ )( 0 m ) = 0

m₂x₂ = non-zero value

c.o.m. = ( m₁x₁ + m₂x₂ )/M

An object that moves with a constant velocity ( v ) travels a fixed number of meters each second:

v = d/t

Thus, the c.o.m. derivation is easily converted to an equation for the c.o.m. velocity ( v_com ) of a system:

v_com = ∑ [ ( m_t )( Δx / t ) ]/M

 v_com = ∑ ( m_t )( v_t ) ]/M

v_com = ∑Δp/M

Prior to analysis of collisions via the use of a moving center of mass, it is useful to see firsthand how the average mass location within the following system remains constant:

The masses are originally separated by a distance of 30 m. As each second transpires, ( m₁ ) will travel ( 3 m ) to the right, and ( m₂ ) will travel ( 7 m ) to the left. After one second transpires, the distance between ( m₁ ) and ( m₂ ) decreases by ( 10 m ):

Δd = 10 m

When two seconds have transpired, ( m₁ ) will have traveled ( 6 m ) to the right, and ( m₂ ) will travel ( 14 m ) to the left. Nonetheless, the distance that separates the masses remains the same:

Δd = 10 m

After three seconds have passed, the masses will meet at the center of mass:

Δd = 10 m

Since the distance separating ( m₁ ) and ( m₂ ) changes at a steady rate, the center of mass shared by these objects must also have a velocity that is constant. As shown, it is not necessary for masses to approach one another with the same velocity in order to share a center of mass. As long as the relative velocity remains the same, so will ( v_cm ).