an artificial satellite of mass m is moving in a circular orbit at a height equal to the radius R of the earth. Suddenly due to internal exploision the satellite breaks into two parts. one part of the satellite stops just after the exploision. the increase in mechanical energy of system due to explosionb will be(acceleration due to gravity is g).

(a)mgR

(b)mgR/2

(c)mgR/4

(d)3mgR/4

mv²/r = GMm/r²                ( v is orbital velocity)

 v² = GM/r       

 v² = GM/2R         ..........1      (r=2R)

satellite is moving initially with velocity v , due to internal forces it explodes so momentam of system must be conserved...

now applying conservation of momentam

mv = mv₁ /2 + mv₂/2                ( one piece comes to rest)

 v₁ = 2v   

since both the particles are at same distance from center of earth so potential energy of system is not changed ,

thus initial P.E is equal to final P.E ...

mecanical energy = PE + KE

 change in mechanical energy is equal to change in KE coz PE is constant

 d(ME) = (KE)_f - (KE)_i

          =1/2 [ (m/2)v₁² - mv²]

          =1/2 [mv²]                            (v₁ = 2v)

now from eq 1 putting v in above eq

d(ME) = GMm/4R

         =mgR/4                               (g = GM/R²)

option c is correct