Question

Mathematically, starting from the discrete 1D center of mass equation, prove that the center of mass velocity of the two carts (Vcom = d/dt Xcom) does not change during an elastic collision.

Answer 1

During elastic collision, momentum is conserved

Let initial momentum of system of two carts be m1v1+m2v2

Velocity of center of mass=vc=(m1v1+m2v2)/(m1+m2)

Let final momentum of system of carts be m1*v1'+m2*v2'

Velocity of center of mass after collision=vc'=(m1*v1'+m2*v2')/(m1+m2)

Since momentum is conserved, m1*v1+m2*v2=m1*v1'+m2*v2'

Therefore, (m1*v1+m2*v2)/(m1+m2)=(m1*v1'+m2*v2')/(m1+m2)

Above equation is obtained by dividing momentum conservation equation by (m1+m2)

Therefore vc=vc'

Therefore, velocity of center of mass is conserved