Question

A satellite of mass 10kg, in a circular orbit around a planet, is having a speed 𝑣=200m/s. The total energy of the satellite is _______kJ. (Rounded off to nearest integer)

Answer 1

Correct Answer: -200

Total Mechanical Energy of a Satellite in Circular Orbit 

The total mechanical energy \( E \) of a satellite in a circular orbit is given by the sum of its kinetic energy and gravitational potential energy:

\[ E = K + U \]

where:

• \( K \) is the kinetic energy of the satellite.
• \( U \) is the gravitational potential energy.

Step 1: Calculate the Kinetic Energy

The kinetic energy \( K \) of the satellite is given by:

\[ K = \frac{1}{2} m v^2 \]

Substituting \( m = 10 \) kg and \( v = 200 \) m/s:

\[ K = \frac{1}{2} \times 10 \times (200)^2 \]

\[ K = 5 \times 40000 = 200000 \text{ J} \]

Step 2: Calculate the Gravitational Potential Energy

The gravitational potential energy \( U \) for an object in orbit is given by:

\[ U = -\frac{GMm}{r} \]

where:

• \( G = 6.67 \times 10^{-11} \) N·m²/kg² is the gravitational constant.
• \( M \) is the mass of the planet (unknown, but can be inferred from the orbital velocity and radius).
• \( r \) is the radius of the orbit (also unknown, but related to the velocity).

For a circular orbit, the gravitational force provides the centripetal force required for the satellite’s circular motion:

\[ \frac{GMm}{r^2} = \frac{m v^2}{r} \]

Solving for \( r \):

\[ r = \frac{GM}{v^2} \]

Substituting this into the potential energy equation:

\[ U = -\frac{1}{2} m v^2 \]

Step 3: Calculate the Total Energy

Thus, the total energy is:

\[ E = K + U = \frac{1}{2} m v^2 - \frac{1}{2} m v^2 = -\frac{1}{2} m v^2 \]

Substituting \( m = 10 \) kg and \( v = 200 \) m/s:

\[ E = -\frac{1}{2} \times 10 \times (200)^2 \]

\[ E = -5 \times 40000 = -200000 \text{ J} \]

\[ E = -200 \text{ kJ} \]

Final Answer

The total energy of the satellite is \(-200\) kJ.