A satellite of mass 500\text{ kg} orbits a spherical planet of radius R = 6.0 \times 10^6\… — JAMB UTME

A satellite of mass $500\text{ kg}$ orbits a spherical planet of radius $R = 6.0 \times 10^6\text{ m}$ with surface gravitational acceleration $g = 10\text{ m/s}^2$ . The satellite is transferred from an initial circular orbit of radius $2R$ to a higher circular orbit of radius $3R$ . What is the minimum energy required, in megajoules ( $\text{MJ}$ ), to perform this transfer?

Cevap: 2500 MJ

Cevap

The minimum energy required to perform the orbital transfer is

2500\text{ MJ}

The minimum energy needed to move a satellite between circular orbits is equal to the change in its total mechanical energy (

E = -\frac{GMm}{2r}

). Expressing

GM

gR^2

, the energy difference between radii

2R

and

3R

simplifies to

\Delta E = \frac{gRm}{12}

, which evaluates to

2500\text{ MJ}

Adım Adım Çözüm

Relate surface acceleration due to gravity to planet mass and radius.

GM = gR^2

At the planet's surface (

r = R

), gravitational acceleration is

g = \frac{GM}{R^2}

Formulate the total mechanical energy equation for a circular orbit.

E = -\frac{GMm}{2r} = -\frac{gR^2 m}{2r}

Total energy is kinetic energy

\frac{GMm}{2r}

plus gravitational potential energy

-\frac{GMm}{r}

Calculate initial and final total energies.

E_1 = -\frac{gRm}{4}

and

E_2 = -\frac{gRm}{6}

Substitute the orbit radii

r_1 = 2R

and

r_2 = 3R

into the total energy equation.

Determine the net work required for the transfer.

\Delta E = E_2 - E_1 = \frac{gRm}{12}

The energy required equals the difference in total mechanical energy between the final and initial orbits.

Substitute given numerical values and convert joules to megajoules.

\Delta E = \frac{10 \times (6.0 \times 10^6) \times 500}{12} = 2.5 \times 10^9\text{ J} = 2500\text{ MJ}

Dividing

2.5 \times 10^9\text{ J}

10^6

converts the value to megajoules.

Anahtar Kavram

Total Mechanical Energy of a Satellite in Circular Orbit and Orbital Transfer Energy

Tahmini Süre:3m 0s

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