No external force: Mass center moving relative to an inertial frame moves with constant velocity
Two particles of masses 2 kg and 3 kg are moving under their mutual interaction in free space. At an instant they were observed at points(-2m,1m,4m) and(2m,-3m,6m) with velocities `(3hati-2hatj+hatk)m//s " and " (-hati+hatj-2hatk)m//s` respectively. If after 10 sec, the first particle passes the point (6m,8m,-6m), find coordinate of the point where the second particle at this instant?

Explanation

System of these two particles is in free, therefore no extenal forces act on them. There total linear momentum remains conserved and their mass center moves with constant velocity relative to an inertial frame.
Velocity of the mass center
`v_c=(Sigmam_ivecv_i)/(Sigmam_i)=(2(3hati-2hatj+hatk)+(3i+j-2hatk))/(2+3) = (3hat(i) - hat(j) - 4 hat(k))/(5) m//s`
Location `vec(r)_(co)`of the mass center at the instant `t = 0 s`

`barr_c=(Sigmam_ibarr_i)/(Sigmami)` rarr " " `barr_infty=(2(-2hati+hatj+4hatk)+3(2hati-3hatj+6hatk))/(2+3)=(2hati-7hatj+26hatk)/(5)`
New location `vec(r )_c` of the mass center at the instant t=10s
`vec(r )_c= vec(r )_infty+vecv_(c) t rarr " " vec(r )_(c) = (2hati-7hatj+26hatk)/(5)=(3i-hatj-4hatk)/5xx10=(32hati-17hatj-14hatk)/5`
New location (x, y, z) of the second particle.
`vec(r)_(c)= (Sigmam_1vecr_1)/(Sigmam_i)rarr` " " `(32hati-17hatj-14hatk)/5=(2(6hati+8hatj-6hatk)+3(xbari+yhatj+zbark))/(2+3)`
Solving the above equation, we obtain the corrdinates of the second particle `(20//3, - 11, -2//3)`