Two satellite fo masses m and 4m orbit the earth in circular orbits of radii 4r and r respectively. The ratio of their orbital speed is

Options

A. 1
B. `1/2`
C. `1/(sqrt2)`
D. `1/(sqrt5)`

Answer

To find the ratio of the orbital speeds of two satellites with masses \( m \) and \( 4m \) orbiting the Earth at radii \( 4r \) and \( r \) respectively, we can follow these steps:

### Step 1: Understand the formula for orbital speed
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 universal gravitational constant, \( M \) is the mass of the Earth, and \( r \) is the radius of the orbit.

### Step 2: Calculate the orbital speed of the first satellite
For the first satellite with mass \( m \) orbiting at radius \( 4r \):
\[ v_1 = \sqrt{\frac{GM}{4r}} = \sqrt{\frac{GM}{r} \cdot \frac{1}{4}} = \frac{1}{2} \sqrt{\frac{GM}{r}} \]

### Step 3: Calculate the orbital speed of the second satellite
For the second satellite with mass \( 4m \) orbiting at radius \( r \):
\[ v_2 = \sqrt{\frac{GM}{r}} \]

### Step 4: Find the ratio of the orbital speeds
Now, we can find the ratio of the orbital speeds \( v_1 \) and \( v_2 \):
\[ \frac{v_1}{v_2} = \frac{\frac{1}{2} \sqrt{\frac{GM}{r}}}{\sqrt{\frac{GM}{r}}} \]
This simplifies to:
\[ \frac{v_1}{v_2} = \frac{1}{2} \]

### Conclusion
Thus, the ratio of the orbital speeds of the two satellites is:
\[ \frac{v_1}{v_2} = \frac{1}{2} \]