Question

Two satellites A and B are moving around the earth in a circular orbit of radius 2R,respectively. If the kinetic energy of the satellite A is two-times the kinetic energy of the satellite B,the ratio of their masses (m_A:m_B) is:

• A. 1:2
• B. 1:1
• C. 2:1
• D. 1:4
• E. 4:1

Answer 1

The Correct Option is C

The kinetic energy (\(K\)) of a satellite in orbit is given by the formula: \[ K = \frac{1}{2} m v^2 \] where \(m\) is the mass of the satellite and \(v\) is the velocity. For a satellite in a circular orbit, the velocity \(v\) is related to the gravitational force providing the centripetal force: \[ \frac{G M m}{R^2} = \frac{m v^2}{R} \] where \(G\) is the gravitational constant, \(M\) is the mass of the Earth, and \(R\) is the radius of the orbit. Solving for \(v^2\): \[ v^2 = \frac{G M}{R} \] Now, the kinetic energy of a satellite becomes: \[ K = \frac{1}{2} m \left( \frac{G M}{R} \right) = \frac{G M m}{2R} \] Let \(K_A\) and \(K_B\) be the kinetic energies of satellites A and B, respectively. According to the given condition: \[ K_A = 2 K_B \] Substituting the expression for kinetic energy: \[ \frac{G M m_A}{2R} = 2 \times \frac{G M m_B}{2R} \] Simplifying: \[ m_A = 2 m_B \] Thus, the ratio of their masses is: \[ m_A : m_B = 2:1 \]

Answer 2

1. Recall the formula for the kinetic energy of a satellite in circular orbit:

The kinetic energy (KE) of a satellite in a circular orbit is given by:

\[KE = \frac{GMm}{2R}\]

where:

• G is the universal gravitational constant
• M is the mass of the Earth
• m is the mass of the satellite
• R is the radius of the orbit

2. Define the given information:

Let KE_A and KE_B be the kinetic energies of satellites A and B, respectively, and m_A and m_B be their masses. Both satellites are in orbits of the same radius R. We are given:

\[KE_A = 2KE_B\]

3. Write the kinetic energy equations for each satellite:

\[KE_A = \frac{GMm_A}{2R}\]

\[KE_B = \frac{GMm_B}{2R}\]

4. Substitute and solve for the mass ratio:

Substitute the given relationship between the kinetic energies:

\[\frac{GMm_A}{2R} = 2\left(\frac{GMm_B}{2R}\right)\]

Simplify and solve for \(m_A/m_B\):

\[\frac{GMm_A}{2R} = \frac{GMm_B}{R}\]

\[m_A = 2m_B\]

\[\frac{m_A}{m_B} = \frac{2}{1}\]