Question

Two masses `nm` and `m`, start simultaneously from the intersection of two straight lines with velocities `v` and `nv` respectively. It is observed that the path of their centre of mass is a straight line bisecting the angle between the given straight lines. Find the magnitude of the velocity of centre of mass. [Here `theta=`angle between the lines]

Answer 1

`vecv_(cm)=(m_(1)vecv_(1)+m_(1)vecv_(2))/(m_(1)+m_(2))`
`vecv_(cm)=(nmvhati+mnvcosthetahati+nmv sin theta hatj)/(m+nm)`
`v_(cm)=sqrt((nmv+mnvcostheta)^(2)+(nmvsintheta)^(2))/(m(1+n))`
`v_(cm)=(nmvsqrt((1+costheta)^(2)+(sintheta)^(2)))/(m(1+n))`
`=(nvsqrt(1+cos^(2)theta+2costheta+sin^(2)theta))/((1+n))`
`=((nv)/(n+1))sqrt(2costheta/2)`
`v_(cm)=(2nv"cos" theta/2)/(n+1)`