To find the orbital speed of a satellite revolving in a circular orbit above the Earth's surface, we can follow these steps:
### Step 1: Identify the given values
- Height of the satellite above the Earth's surface, \( h = 0.25 \times 10^6 \, m \)
- Radius of the Earth, \( R = 6.38 \times 10^6 \, m \)
- Acceleration due to gravity at the surface of the Earth, \( g = 9.8 \, m/s^2 \)
### Step 2: Calculate the distance from the center of the Earth to the satellite
The total distance \( r \) from the center of the Earth to the satellite is given by:
\[
r = R + h
\]
Substituting the values:
\[
r = 6.38 \times 10^6 \, m + 0.25 \times 10^6 \, m = 6.63 \times 10^6 \, m
\]
### Step 3: Use the formula for orbital speed
The formula for the orbital speed \( v \) of a satellite in a circular orbit is given by:
\[
v = \sqrt{\frac{GM}{r}}
\]
Where \( G \) is the gravitational constant. However, we can also express \( GM \) in terms of \( g \) and \( R \):
\[
GM = gR^2
\]
Thus, the orbital speed can be rewritten as:
\[
v = \sqrt{\frac{gR^2}{r}}
\]
### Step 4: Substitute the known values into the formula
Now, substituting \( g = 9.8 \, m/s^2 \), \( R = 6.38 \times 10^6 \, m \), and \( r = 6.63 \times 10^6 \, m \):
\[
v = \sqrt{\frac{9.8 \times (6.38 \times 10^6)^2}{6.63 \times 10^6}}
\]
### Step 5: Calculate the value
First, calculate \( (6.38 \times 10^6)^2 \):
\[
(6.38 \times 10^6)^2 = 4.065044 \times 10^{13} \, m^2
\]
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:
\[
9.8 \times 4.065044 \times 10^{13} \approx 3.98 \times 10^{14}
\]
Now divide by \( 6.63 \times 10^6 \):
\[
\frac{3.98 \times 10^{14}}{6.63 \times 10^6} \approx 5.99 \times 10^7
\]
Finally, take the square root:
\[
v \approx \sqrt{5.99 \times 10^7} \approx 7.76 \times 10^3 \, m/s
\]
### Step 6: Convert to kilometers per second
To convert \( m/s \) to \( km/s \):
\[
v \approx 7.76 \, km/s
\]
### Final Answer
The orbital speed of the satellite is approximately \( 7.76 \, km/s \).
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