Motion of Mass Center in Vector Form
A 2.0 kg particle has a velocity of Vecv_1=(2.0hati-3.0hatj)m /s, and a 3.0 kg particle has a velocity vecv_2=(1.0hati+6.0hatj)m/s.
(a) How fast is the center of mass of the particle system moving?
(b) Find velocities of both the particles in centroidal frame.

Explanation

(a) Velocity of the mass center `vec(v)_(c ) = (m_(1)vec(v)_(1) + m_(2)vec(v)_(2))/(m_(1)+m_(2))`
`vec(v)_(c ) = (m_(1)vec(v)_(1)+m_(2)vec(v)_(2))/(m_(1)+m_(2)) rarr vec(v)_(c ) = (2 (2.0hat(i) - 3.0hat(j)) + 3 (1.0hat(i) + 6.0 hat(j)))/(2+3)=(1.4hat(i)+2.4hat(j))m//s`
(b) Velocity of the first particle in centrodial frame
`vec(v)_(1//c) = vec(v)_(1) - vec(v)_(c ) rarr " " vec(v)_(2//c) = (2.0 hat(i) - 3.0 hat(j) - (1.4hat(j)) = 0.6 (hat(i) + hat(j)) m//s`
Velocity of the second particle in centrodial frame
`vec(v)_(2//c) = vec(v)_(2) - vec(c ) rarr " " vec(v)_(2//c) = (1.0 hat(j)) - (1.4 hat(i) + 2.4 hat(j)) = - 0.4 hat(i) + 3.6 hat(j) m//s`