Two particles of masses m, and m, separated by a distance dare 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 fest, so the velocity of the centre of mass,
`vecv_(CM)=(m_1xx0+m_2xx0)/(m_1+m_2)=0, ` As here `vecF_("ext")= 0 ` so , `vecv_(CM)` = constant
ie, `(m_1vecv_1+m_2vecv_2)/(m_1+m_2)=0` (at all instants) or `m_1 +vecm_1vecv_1 +m_2vecv_2 = 0`
or `m_t(Deltavecr_1)/(Deltat)+m_2(Deltavecr_2)/(Deltat)=0`
`m_1Deltavecr_1+m_2Deltavecr_2=0 " " (asDeltatneoo)`
`m_1vecd_1+m_2vecd_2=0" "` (with `Deltavecr=vecd`)
`m_1d_1-m_2d_2=0` (as direction `vecd_2` is opposite to `d_1` ) or `m_1d_1=m_2d_2`
But given that `d_1+d_2=d` so that `d_1=(m_2d)/(m_1+m_2)and d_2=(m_1d)/(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,