Question

Two satellites \( A \) and \( B \) are revolving with critical velocities \( v_A \) and \( v_B \) around the earth, in circular orbits of radii \( R \) and \( 2R \), respectively. The ratio \( \frac{v_A}{v_B} \) is

• A. \( 2 : 1 \)
• B. \( \sqrt{2} : 1 \)
• C. \( 1 : 2 \)
• D. \( 1 : \sqrt{2} \)

Hint:

The critical velocity for a satellite in orbit is inversely proportional to the square root of the radius. When comparing two orbits, use the ratio of the radii to find the ratio of the velocities.

Answer 1

The Correct Option is B

Step 1: Understanding the critical velocity formula.
The critical velocity \( v \) of a satellite in circular orbit is given by the formula: \[ v = \sqrt{\frac{GM}{R}} \] where \( G \) is the gravitational constant, \( M \) is the mass of the earth, and \( R \) is the radius of the orbit.
Step 2: Using the formula to find the ratio of velocities.
For satellites \( A \) and \( B \), the critical velocities are given by: \[ v_A = \sqrt{\frac{GM}{R}} \quad \text{and} \quad v_B = \sqrt{\frac{GM}{2R}} \] Taking the ratio \( \frac{v_A}{v_B} \), we get: \[ \frac{v_A}{v_B} = \frac{\sqrt{\frac{GM}{R}}}{\sqrt{\frac{GM}{2R}}} = \sqrt{2} \]
Step 3: Conclusion.
Thus, the ratio of \( v_A \) to \( v_B \) is \( \sqrt{2} : 1 \), which corresponds to option (B).