An earth satellite makes a complete revolution around the earth in 120 minutes. If the orbit is circular calculate the height of satellite above the earth. Radius of the earth = 6400 km g = `9.8 ms^(-2)` .

Explanation

To solve the problem of finding the height of a satellite above the Earth when it completes a revolution in 120 minutes, we can follow these steps:

### Step 1: Convert the time period into seconds
The time period \( T \) is given as 120 minutes. We need to convert this into seconds.

\[ T = 120 \text{ minutes} \times 60 \text{ seconds/minute} = 7200 \text{ seconds} \]

**Hint:** Remember to convert minutes to seconds by multiplying by 60.

### Step 2: Use the formula for the orbital radius
According to Kepler's third law, the square of the time period \( T \) is proportional to the cube of the radius \( R \) of the orbit. The formula can be expressed as:

\[ T^2 = \frac{4\pi^2}{g} R^3 \]

Where:
- \( g \) is the acceleration due to gravity (9.8 m/s²)
- \( R \) is the distance from the center of the Earth to the satellite.

Rearranging the formula gives:

\[ R = \left(\frac{g T^2}{4\pi^2}\right)^{1/3} \]

**Hint:** Make sure to isolate \( R \) in the equation to find the orbital radius.

### Step 3: Substitute the known values
Now we can substitute the known values into the formula:

- \( g = 9.8 \, \text{m/s}^2 \)
- \( T = 7200 \, \text{s} \)

\[ R = \left(\frac{9.8 \times (7200)^2}{4\pi^2}\right)^{1/3} \]

### Step 4: Calculate \( R \)
First, calculate \( T^2 \):

\[ T^2 = 7200^2 = 51840000 \, \text{s}^2 \]

Now substitute \( T^2 \) into the equation:

\[ R = \left(\frac{9.8 \times 51840000}{4 \times (3.14)^2}\right)^{1/3} \]

Calculating the denominator:

\[ 4 \times (3.14)^2 \approx 39.4784 \]

Now calculate the entire expression:

\[ R = \left(\frac{9.8 \times 51840000}{39.4784}\right)^{1/3} \]

Calculating the numerator:

\[ 9.8 \times 51840000 \approx 508032000 \]

Now divide by the denominator:

\[ \frac{508032000}{39.4784} \approx 12850.4 \]

Now take the cube root:

\[ R \approx (12850.4)^{1/3} \approx 23.0 \times 10^3 \, \text{m} \approx 23000 \, \text{m} \]

### Step 5: Calculate the height above the Earth's surface
The height \( h \) of the satellite above the Earth's surface can be found by subtracting the Earth's radius from \( R \):

\[ h = R - R_e \]

Where \( R_e = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6400000 \, \text{m} \).

Now substitute \( R \) and \( R_e \):

\[ h = 23000 - 6400000 \]

Calculating \( h \):

\[ h \approx 23000 - 6400000 \approx 1681 \times 10^3 \, \text{m} = 1681 \, \text{km} \]

### Final Answer
The height of the satellite above the Earth's surface is approximately **1681 km**.

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