Two satellites of masses 80 kg and 120 kg revolve round a planet in circular orbits of radii 16 R and 9 R respectively, where R is radius of the planet. The ratio of the speeds of satellites will be

Options

A. `(80)/(120)`
B. `(4)/(3)`
C. `(16)/(25)`
D. `(3)/(4)`

Explanation

To find the ratio of the speeds of two satellites revolving around a planet, we can use the formula for orbital velocity. The orbital velocity \( V \) of a satellite in a circular orbit is given by:

\[ V = \sqrt{\frac{GM}{R}} \]

where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the distance from the center of the planet to the satellite (which is the radius of the orbit).

In this case, we have two satellites:
1. Satellite 1 (mass = 80 kg) at radius \( R_1 = 16R \)
2. Satellite 2 (mass = 120 kg) at radius \( R_2 = 9R \)

### Step 1: Calculate the orbital velocity for Satellite 1

Using the formula for orbital velocity:

\[ V_1 = \sqrt{\frac{GM}{R_1}} = \sqrt{\frac{GM}{16R}} \]

### Step 2: Calculate the orbital velocity for Satellite 2

Similarly, for Satellite 2:

\[ V_2 = \sqrt{\frac{GM}{R_2}} = \sqrt{\frac{GM}{9R}} \]

### Step 3: Find the ratio of the speeds of the satellites

Now, we want to find the ratio \( \frac{V_1}{V_2} \):

\[ \frac{V_1}{V_2} = \frac{\sqrt{\frac{GM}{16R}}}{\sqrt{\frac{GM}{9R}}} \]

### Step 4: Simplify the ratio

The \( GM \) and \( R \) terms cancel out:

\[ \frac{V_1}{V_2} = \frac{\sqrt{1/16}}{\sqrt{1/9}} = \frac{\frac{1}{4}}{\frac{1}{3}} = \frac{3}{4} \]

### Conclusion

Thus, the ratio of the speeds of the satellites is:

\[ \frac{V_1}{V_2} = \frac{3}{4} \]