Two particles of masses `m_(1) and m_(2)` separated by a distance d are at rest initially. If they move towards each other under mutual interaction (say electric, gravitational or elastic), where will they meet?

Options

A. At the centre of the joining the two particles
B. Anywhere in between two masses
C. At the centre of mass of the system of two particles
D. none of the above

Answer

C

Explanation

Initially, the particles are at rest, so the velocity of the centre of mass,
`vec(v)_(CM)=(m_(1) times 0+ m_(2) times0)/(m_(1)+m_(2))=0," "` As here `vec(F)_(ext)=0,"so "vec(v)_(CM)="constant"`
ie, `(m_(1)vec(v)_(1)+m_(2)vec(v)_(2))/(m_(1)+m_(2))=0" (at all instants) "or m_(1)vec(v)_(1)+m_(2)vec(v)_(2)=0`
or `m_(1)(Deltavec(r)_(1))/(Deltat)+m_(2)(Deltavec(r)_(2))/(Deltat)=0`
`m_(1)Deltavec_(r)_(1)+m_(2)Deltavec(r)_(2)=0" "("as "Deltat ne infty)`
`m_(1)vec(d)_(1)+m_(2)vec(d)_(2)=0" "("with "Deltavec(r)=vec(d))`
`m_(1)d_(1)=m_(2)d_(2)=0` (as direction `vec(d)_(2)` is opposite to `d_(1)`) `" "or m_(1)d_(1)=m_(2)d_(2)`
But given that `d_(1)+d_(2)=d`, so that `d_(1)=(m_(2)d)/(m_(1)+m_(2)) and d_(2)=(m_(1)d)/(m_(1)+m_(2))`
Now, as `d_(1) and d_(2)` represent the position of the centre of the mass relative to `m_(1) and m_(2)` respectively the particles will collide at the centre of mass of the system.