Question

A 1480 kg satellite orbits the Earth at an altitude of 61.6 km. It is desired to increase the altitude of the orbit to 285 km. How much energy must be added to the system to effect this change in altitude? The universal gravitational constant is 6.67259 × 10^−11 N · m2/kg2 , the radius of the Earth is 6.37×10^6 m, and its mass is 5.98×10^24 kg.

Answer 1

The change in work done on the satellite gives the energy to be added to the system.

i.e $\Delta$W = $\Delta$F. $\Delta$x

   $\Delta$W = change in work done , $\Delta$F = change inforce of attraction between the satellite and the earth

   $\Delta$x = change in the altitudes = h_f - h_i = 285000 m - 61600 m

   = 223400 m

G = universal gravitational constant = 6.67259 x 10^-11 Nm²/Kg²

m_s = mass of satellite = 1480 Kg , M_e = mass of the earth = 5.98 x10²⁴ Kg

r_i = initial distance between the centres of the satellite and the earth

= ( R + h_i ) here R = radius of the earth=6.37 x10⁶ m , h_{i = initial altitude = 61.6 Km = 61600 m}

  = ( 6370000 m + 61600 m )

   = 6431600 m

r_f = final distance between the centres of the satellite and the earth

   = ( R + h_f ) here h_f = final altitude = 285 Km = 285000 m

   = ( 6370000 m + 285000 m )

   = 6655000 m

Now the work done = the change in kinetic energy

  $\Delta$ W = G m_s M_e [ 1/r_f²   - 1/r_i² ] $\Delta$x

   = 6.67 x10^-11 x 1480 x5.98 x10²⁴ [ 1 / (6655000)²   - 1/ ( 6431600 )²  ] 223400

   = - 2.0685343542 x10⁸ J here '-' indicates against to the gravity

   = 2.07 x10⁸ J ( approx. )

So, 2.07 x10⁸ J energy should be added to the system .