Two particles of masses 2 kg and 4 kg are approaching each other with acceleration `1 ms^(-2)` and `2 ms^(-2)`, respectively, on a smooth horizontal surface. Find the acceleration of center of mass of the system.

Explanation

The acceleration of c.m. of the system is given by
`vec(a)_(cm)=(m_(1)vec(a)_(1)+m_(2)vec(a)_(2))/(m_(1)+m_(2))`
`rArr a_(cm)=(|m_(1)vec(a)_(1)+m_(2)vec(a)_(2)|)/(m_(1)+m_(2))`
Since `vec(a)_(1)` & `vec(a)_(2)` are anti - parallel, `a_(cm)=(|m_(1)a_(1)-m_(2)a_(2)|)/(m_(1)+m_(2))`
`rArr a_(cm)=(|(2)(1)-(4)(2)|)/(2+4)m//sec^(2)=1m//sec^(2)`
Since `m_(2)a_(2)gt m_(1)a_(1)`, the direction of acceleration of c.m. will be directed in the direction of `a_(2)`.