Question

Two satellites A and B go around a planet in circular orbits having radii of 4R and R, respectively. If the velocity of satellite A is 3v, the velocity of satellite B will be:

• A. 18v
• B. 12v
• C. 6v
• D. 3v

Hint:

Remember the relationships for orbital motion: - Velocity: \( v \propto 1/\sqrt{r} \) (closer satellite is faster) - Period: \( T \propto r^{3/2} \) (closer satellite has shorter period)

Answer 1

The Correct Option is C

Step 1: State the formula for the orbital velocity of a satellite. The velocity (\(v\)) of a satellite in a circular orbit of radius \(r\) around a planet of mass \(M\) is given by equating the gravitational force to the centripetal force: \[ \frac{GMm}{r^2} = \frac{mv^2}{r} $\Rightarrow$ v = \sqrt{\frac{GM}{r}} \] This shows that the orbital velocity is inversely proportional to the square root of the orbital radius: \( v \propto \frac{1}{\sqrt{r}} \).

Step 2: Set up a ratio for the two satellites A and B. \[ \frac{v_B}{v_A} = \frac{\sqrt{GM/r_B}}{\sqrt{GM/r_A}} = \sqrt{\frac{r_A}{r_B}} \]

Step 3: Substitute the given values and solve for \(v_B\). - \( v_A = 3v \) - \( r_A = 4R \) - \( r_B = R \) \[ \frac{v_B}{3v} = \sqrt{\frac{4R}{R}} = \sqrt{4} = 2 \] \[ v_B = 2 \times (3v) = 6v \]