Orbital speed of a satellite

Options

A. decreases with increase in height of satellite.
B. is independent of the mass of satellite.
C. is along the tangent to the orbital path of satellite.
D. is independent of the mass and radius of the planet around which the revolution of satellite takes place.

Answer

To find the orbital speed of a satellite, we need to derive the formula based on the forces acting on the satellite. Here’s a step-by-step solution:

### Step 1: Understanding the Forces
A satellite in orbit around a planet experiences gravitational force, which provides the necessary centripetal force for its circular motion. The gravitational force acting on the satellite is given by Newton's law of gravitation.

### Step 2: Write the Gravitational Force Equation
The gravitational force \( F_g \) acting on the satellite can be expressed as:
\[ F_g = \frac{G \cdot M \cdot m}{r^2} \]
where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the planet,
- \( m \) is the mass of the satellite,
- \( r \) is the distance from the center of the planet to the satellite.

### Step 3: Write the Centripetal Force Equation
For an object in uniform circular motion, the centripetal force \( F_c \) required to keep the satellite in orbit is given by:
\[ F_c = \frac{m \cdot v^2}{r} \]
where:
- \( v \) is the orbital speed of the satellite.

### Step 4: Set the Forces Equal
Since the gravitational force provides the centripetal force, we can set these two equations equal to each other:
\[ \frac{G \cdot M \cdot m}{r^2} = \frac{m \cdot v^2}{r} \]

### Step 5: Cancel Out the Mass of the Satellite
We can cancel the mass of the satellite \( m \) from both sides of the equation (assuming \( m \neq 0 \)):
\[ \frac{G \cdot M}{r^2} = \frac{v^2}{r} \]

### Step 6: Solve for Orbital Speed
Now, multiply both sides by \( r \) to isolate \( v^2 \):
\[ \frac{G \cdot M}{r} = v^2 \]
Taking the square root of both sides gives us the formula for the orbital speed \( v \):
\[ v = \sqrt{\frac{G \cdot M}{r}} \]

### Step 7: Analyze the Result
From the derived formula \( v = \sqrt{\frac{G \cdot M}{r}} \), we can analyze how the orbital speed depends on various factors:
1. **Height of the Satellite**: As the height increases (which increases \( r \)), the orbital speed \( v \) decreases.
2. **Mass of the Satellite**: The mass \( m \) of the satellite does not appear in the final formula, indicating that the orbital speed is independent of the satellite's mass.
3. **Mass of the Planet**: The orbital speed depends on the mass \( M \) of the planet.
4. **Radius**: The orbital speed depends on the radius \( r \) from the center of the planet.

### Conclusion
The orbital speed of a satellite decreases with an increase in height, is independent of the mass of the satellite, and is dependent on the mass of the planet and the radius of the orbit.

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