Question

Recall that the blocks can only move along the x axis. the x components of their velocities at a certain moment are v1x and v2x. find the x component of the velocity of the center of mass (vcm)x at that moment. keep in mind that, in general: vx=dx/dt. express your answer in terms of m1, m2, v1x, and v2x.

Answer 1

The center of mass is given with this formula:
$x_c=\frac{\sum_{n=1}^{n=i}m_ix_i}{M}$
Velocity is:
$v=\frac{dv}{dt}$
So, for the velocity of the center of mass we have:
$\frac{dx_c}{dt}=\frac{\sum_{n=1}^{n=i}d(m_ix_i)}{Mdt}\\ v_c=\frac{\sum_{n=1}^{n=i}p_i}{M}$
In our case it is:
$v_{xc}=\frac{m_1v_{x1}+m_2v_{x2}}{m_1+m_2}$