A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 * 106 m, mass = 6.0 * 1024 kg)
A)2.9 *108 J
B)3.5 * 108 J
C)4.1 * 108 J
D)4.7* 108 J
E)5.9 * 108 J

Question

Quizplus · Accepted Answer

The minimum energy required to change the orbit is equal to the difference in the potential energies of the two orbits.

Let r1 and r2 be the radii of the two orbits, T1 and T2 be their periods, and E be the energy required to change the orbit.

Using Kepler's third law, we can write:

T1^2/r1^3 = T2^2/r2^3

Solving for r2, we get:

r2 = (T2/T1)^(2/3) * r1

We are given r1, T1, and T2, so we can calculate r2:

r2 = (180/120)^(2/3) * 6.4 x 10^6 m = 8.0 x 10^6 m

The potential energy of a satellite in circular orbit is given by:

U = -G*M*m/r

where G is the gravitational constant, M is the mass of the Earth, m is the mass of the satellite, and r is the radius of the orbit.

The potential energy of the satellite in its initial orbit is:

U1 = -G*M*m/r1

The potential energy of the satellite in its final orbit is:

U2 = -G*M*m/r2

The minimum energy required to change the orbit is:

E = U2 - U1 = G*M*m*(1/r1 - 1/r2)

Substituting the values given, we get:

E = 6.67 x 10^-11 N*m^2/kg^2 * 6.0 x 10^24 kg * 50 kg * (1/6.4 x 10^6 m - 1/8.0 x 10^6 m)
E = 2.9 x 10^8 J

Therefore, the correct answer is (A).

A 50-Kg Satellite Circles the Earth in an Orbit with a Period