Calculate the orbital speed of the Hubble space telescope orbiting at a height of 598 km above the earth's surface. Take R = 6400km. Mass of earth `- 5.98 xx 10^(24) `kg

Explanation

To calculate the orbital speed of the Hubble Space Telescope orbiting at a height of 598 km above the Earth's surface, we can follow these steps:

### Step 1: Identify the given values
- Height of the Hubble Space Telescope above the Earth's surface, \( h = 598 \) km
- Radius of the Earth, \( R = 6400 \) km
- Mass of the Earth, \( M = 5.98 \times 10^{24} \) kg
- Universal gravitational constant, \( G = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \)

### Step 2: Convert the height and radius into meters
Since the height and radius are given in kilometers, we need to convert them into meters:
- Height in meters:
\[ h = 598 \, \text{km} = 598 \times 10^3 \, \text{m} = 598000 \, \text{m} \]
- Radius of the Earth in meters:
\[ R = 6400 \, \text{km} = 6400 \times 10^3 \, \text{m} = 6400000 \, \text{m} \]

### Step 3: Calculate the total distance from the center of the Earth to the satellite
The total distance \( r \) from the center of the Earth to the Hubble Space Telescope is the sum of the Earth's radius and the height of the telescope:
\[ r = R + h = 6400000 \, \text{m} + 598000 \, \text{m} = 6998000 \, \text{m} \]

### Step 4: Use the formula for orbital speed
The formula for the orbital speed \( v \) of a satellite is given by:
\[ v = \sqrt{\frac{GM}{r}} \]
Substituting the values of \( G \), \( M \), and \( r \):
\[ v = \sqrt{\frac{(6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2)(5.98 \times 10^{24} \, \text{kg})}{6998000 \, \text{m}}} \]

### Step 5: Calculate the value inside the square root
First, calculate \( GM \):
\[ GM = (6.67 \times 10^{-11})(5.98 \times 10^{24}) \approx 3.986 \times 10^{14} \, \text{m}^3/\text{s}^2 \]
Now, substitute this value into the formula:
\[ v = \sqrt{\frac{3.986 \times 10^{14}}{6998000}} \approx \sqrt{5.698 \times 10^7} \approx 7545.6 \, \text{m/s} \]

### Step 6: Final answer
Thus, the orbital speed of the Hubble Space Telescope is approximately:
\[ v \approx 7.55 \times 10^3 \, \text{m/s} \]