Two satellites are orbiting the earth in circular orbits of radii in the ratio 4: 1. The ratio of their orbital speed is

Question

Accepted Answer

To determine the ratio of the orbital speeds of two satellites orbiting the Earth in circular orbits with radii in the ratio 4:1, we need to consider the relationship between orbital speed and the radius of the orbit.

1. **Explanation for Option 4 (1:2):**

The orbital speed $ v $ of a satellite in a 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.

Given that the radii of the orbits are in the ratio 4:1, let's denote the larger radius as $ 4r $ and the smaller radius as $ r $.

- For the satellite in the larger orbit (radius $ 4r $):
     $$
     v_1 = \sqrt{\frac{GM}{4r}} = \frac{1}{2} \sqrt{\frac{GM}{r}}
     $$

- For the satellite in the smaller orbit (radius $ r $):
     $$
     v_2 = \sqrt{\frac{GM}{r}}
     $$

The ratio of their orbital speeds is:
   $$
   \frac{v_1}{v_2} = \frac{\frac{1}{2} \sqrt{\frac{GM}{r}}}{\sqrt{\frac{GM}{r}}} = \frac{1}{2}
   $$

Therefore, the ratio of their orbital speeds is 1:2, which corresponds to Option 4.

2. **Explanation for why other options are incorrect:**

- **Option 1 (16:1):** This option suggests that the orbital speed ratio is 16:1. However, since the orbital speed is inversely proportional to the square root of the radius, such a large ratio is not possible given the radius ratio of 4:1.

- **Option 2 (Sat Dec 30 1899 14:37:10 GMT+0521 (GMT+05:30)):** This is not a valid numerical ratio and does not relate to the problem of calculating orbital speeds.

- **Option 3 (2:1):** This option suggests that the orbital speed of the satellite in the larger orbit is twice that of the smaller orbit. However, as derived above, the speed decreases with an increase in radius, leading to a ratio of 1:2, not 2:1.

In conclusion, Option 4 (1:2) is the correct answer, as it accurately reflects the relationship between the orbital speeds and the radii of the orbits.