Question

A satellite is moving Only a circular orbit around the earth Again a height 6400km if the satellite Is suddelny stopped in its orbit and allowed to fall freely towards earth find the speed with which it hits the surface of the earth

Answer 1

Height of the satellite from the surface of Earth = H = 6400 km.
Radius of Earth = R = 6400 km.
Mass of Earth = M.
G = Universal Gravitational constant.

Initial KE = 0.  Speed = 0.    
Gravitational potential energy of the satellite at height 6400 km 
   PE1 = - G M m/(H+R) = - G M m/ (2R) 
           = - (G M/R²) * (mR/2)
          = - m g R / 2         ,    as gravity g on Earth's surface = GM/R²

This potential energy on the surface of Earth:
     PE2 = - GMm/R = - (GM/R²) * (mR)
             = - m g R

Potential Energy lost = PE1 - PE2
       ΔPE  = m g R/2

Kinetic energy gained = ΔKE = 1/2 m v²
              1/2 m v² = m g R/2
               v² = g R
                v = √[g R] 
                   = √[9.81 * 6400,000]   m/s
                v = 7923.64 m/s. = 28, 525 km/hr