A remote-sensing satellite of earth revolves in a circular orbit at a hight of `0.25xx10^(6)m` above the surface of earth. If earth's radius is `6.38xx10^(6)m` and `g=9.8ms^(-2)`, then the orbital speed of the satellite is

Options

A. `6.67kms^(-1)`
B. `7.76kms^(-1)`
C. `8.56kms^(-1)`
D. `9.13kms^(-1)`

Explanation

To find the orbital speed of a satellite revolving in a circular orbit at a height of \(0.25 \times 10^6 \, m\) above the Earth's surface, we can follow these steps:

### Step 1: Determine the total radius of the orbit
The total radius \(R\) of the orbit is the sum of the Earth's radius \(R_e\) and the height \(h\) of the satellite above the Earth's surface.

\[ R = R_e + h \]

Given:
- \(R_e = 6.38 \times 10^6 \, m\)
- \(h = 0.25 \times 10^6 \, m\)

Calculating \(R\):

\[ R = 6.38 \times 10^6 \, m + 0.25 \times 10^6 \, m = 6.63 \times 10^6 \, m \]

### Step 2: Use the formula for orbital speed
The orbital speed \(v\) of a satellite can be derived from the balance of gravitational force and centripetal force. The formula for orbital speed is given by:

\[ v = \sqrt{\frac{g R_e^2}{R}} \]

Where:
- \(g = 9.8 \, m/s^2\) (acceleration due to gravity at the Earth's surface)
- \(R_e\) is the radius of the Earth
- \(R\) is the total radius of the orbit calculated in Step 1.

### Step 3: Substitute the values into the formula
Substituting the known values into the formula:

\[ v = \sqrt{\frac{9.8 \times (6.38 \times 10^6)^2}{6.63 \times 10^6}} \]

### Step 4: Calculate the value inside the square root
First, calculate \(R_e^2\):

\[ R_e^2 = (6.38 \times 10^6)^2 = 4.065044 \times 10^{13} \, m^2 \]

Now substitute this back into the equation:

\[ v = \sqrt{\frac{9.8 \times 4.065044 \times 10^{13}}{6.63 \times 10^6}} \]

Calculating the numerator:

\[ 9.8 \times 4.065044 \times 10^{13} = 3.98 \times 10^{14} \]

Now divide by \(6.63 \times 10^6\):

\[ \frac{3.98 \times 10^{14}}{6.63 \times 10^6} \approx 5.99 \times 10^7 \]

### Step 5: Take the square root
Now take the square root:

\[ v \approx \sqrt{5.99 \times 10^7} \approx 7.74 \times 10^3 \, m/s \]

### Step 6: Convert to kilometers per second
To convert from meters per second to kilometers per second:

\[ v \approx 7.74 \, km/s \]

### Final Answer
The orbital speed of the satellite is approximately \(7.74 \, km/s\).

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