Satellites `A` and `B` are orbiting around the earth in orbits of ratio `R` and `4R` respectively. The ratio of their areal velocities is-

Options

A. `1:2`
B. `1:4`
C. `1:8`
D. `1:16`

Explanation

To find the ratio of the areal velocities of satellites A and B, we can follow these steps:

### Step 1: Understand Areal Velocity
Areal velocity (dA/dt) is defined as the area swept out by the radius vector of a satellite per unit time. According to Kepler's second law, the areal velocity is constant for a satellite in orbit.

### Step 2: Use the Formula for Areal Velocity
From the video transcript, we know that the areal velocity (dA/dt) is directly proportional to the square root of the radius (R) of the orbit:
\[ \frac{dA}{dt} \propto \sqrt{R} \]

### Step 3: Define the Radii of the Orbits
Let:
- Radius of satellite A = R
- Radius of satellite B = 4R

### Step 4: Write the Areal Velocities
Using the proportionality:
- Areal velocity of satellite A:
\[ \frac{dA}{dt}_A \propto \sqrt{R} \]
- Areal velocity of satellite B:
\[ \frac{dA}{dt}_B \propto \sqrt{4R} = 2\sqrt{R} \]

### Step 5: Find the Ratio of Areal Velocities
Now, we can find the ratio of the areal velocities of satellites A and B:
\[ \frac{dA/dt_A}{dA/dt_B} = \frac{\sqrt{R}}{2\sqrt{R}} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the areal velocities of satellites A and B is:
\[ \frac{dA/dt_A}{dA/dt_B} = \frac{1}{2} \]