# Two objects of masses m and 4m are at rest at an infinite separation. They move towards each other under mutual gravitational attraction. If G i

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Two objects of masses m and 4m are at rest at an infinite separation. They move towards each other under mutual gravitational attraction. If G is the universal gravitaitonal constant, then at separation r
A. the total mechanical energy of the two objects is zero
B. their relative velocity is sqrt((10Gm)/(r))
C. the total kinetic energy of the object is (4Gm^(2))/(r)
D. their relative velocity is zero.

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By applying low of conservation of momentum
mv_(1)-4mv_(2)=0impliesv_(1)=4v_(2)
by applying conservation of energy (1)/(2)mv_(1)^(2)+(1)/(2)4mv_(2)^(2)=(Gm4m)/(r)implies10mv_(2)^(2)=(G4m^(2))/(r)impliesv_(2)=2sqrt((Gm)/(10r))
therefore total kinetic energy =(4Gm^(2))/(r)
Relative velocity for the particle impliesv_(rel)=|vecv_(1)-vecv_(2)|=5v_(2)=sqrt((10Gm)/(r))
Mechanical energy of system =0= constant by using reduced mass concept
(1)/(2)muv_(rel)^(2)=(Gm(4m))/(r) where mu=((m)(4m))/(m+4m)=(4)/(5)mimpliesv_(rel)=sqrt((10Gm)/(r))
Also total KE system =(G(m)(4m))/(r)=(4Gm^(2))/(r)