Problem 55 - Algebra 2

Question

Satellites The circular velocity \(v,\) in miles per hour, of a satellite orbiting Earth is given by the formula \(v=\sqrt{\frac{1.24 \times 10^{12}}{r}, \text { where } r \text { is the distance }}\) in miles from the satellite to the center of Earth. How much greater is the velocity of a satellite orbiting at an altitude of 100 \(\mathrm{mi}\) than one orbiting at an altitude of 200 \(\mathrm{mi} ?\) (The radius of Earth is 3950 \(\mathrm{mi}\) .)

Step-by-Step Solution

Answer

To find the difference in velocity between the two altitudes, first calculate the velocities at those altitudes using the given velocity formula, then subtract the two speeds.

Step 1: Determine the total distance from the center of the Earth for each altitude

The total distance 'r' from the center of the earth would be the sum of the Earth's radius (3950 miles) and the altitude of the satellite. For an altitude of 100 miles, \(r_1 = 3950 + 100 = 4050\) miles. Similarly, for an altitude of 200 miles, \(r_2 = 3950 + 200 = 4150\) miles.

Step 2: Calculate the velocities at each altitude

Use the formula for the velocity of a satellite, \(v = \sqrt{\frac{1.24 \times 10^{12}}{r}}\), and substitute \(r_1\) and \(r_2\) into the equation. This gives \(v_1 = \sqrt{\frac{1.24 \times 10^{12}}{4050}}\) and \(v_2 = \sqrt{\frac{1.24 \times 10^{12}}{4150}}\). Compute these values to get the velocities.

Step 3: Determine the difference in velocities

Subtract the velocity at 4150 miles (\(v_2)\) from the velocity at 4050 miles (\(v_1)\) to find the difference. This can be calculated as \(v = v_1 - v_2\).

Key Concepts

Satellite Orbit

A satellite orbit refers to the path a satellite takes as it moves around Earth. Satellites can orbit at various altitudes, each affecting their speed and function. Orbits can be defined by their shape, commonly circular or elliptical. For most calculations involving circular orbits, the key variable is the radius of the orbit.

• Orbits closer to the Earth are influenced more strongly by Earth's gravitational pull, resulting in higher velocities.
• Satellites with circular orbits maintain a constant distance from the Earth's surface.
• High-altitude orbits, such as those used for geostationary satellites, allow coverage of larger areas.

Understanding satellite orbits helps in predicting satellite behavior, calculating precise orbital periods, and ensuring the satellite serves its intended purpose effectively.

Distance from Earth's Center

When calculating the velocity of a satellite in orbit, the distance from the Earth's center is crucial. This distance not only includes the altitude of the satellite above the Earth's surface but also the Earth's radius itself. This total distance is vital for understanding how quickly a satellite needs to move to maintain its orbit.

• The radius of Earth is approximately 3950 miles.
• Total distance 'r' involves adding the satellite’s altitude to Earth's radius.
• For instance, a satellite at a 100-mile altitude calculates as a total distance of 4050 miles from the center of Earth.
• Similarly, a satellite at a 200-mile altitude results in a distance of 4150 miles.

Knowing these distances helps in using the formula for circular velocity correctly, which impacts the satellite's stability in its orbit.

Velocity Calculation

The circular velocity formula for a satellite helps us calculate the speed needed to maintain its orbit around Earth. This is achieved using the formula:
\[v = \sqrt{\frac{1.24 \times 10^{12}}{r}} \]where \( v \) is the velocity, and \( r \) is the distance from Earth's center.

• The equation arises from balancing gravitational pull with the satellite's centripetal force.
• As \( r \) increases, the value under the square root sign decreases, producing a lower velocity \( v \).
• For example, a satellite at 4050 miles gives a different velocity compared to one at 4150 miles, as demonstrated in calculations for altitudes of 100 miles and 200 miles, respectively.

By understanding and applying this formula, students can discern the changes in velocity required for satellites at varying altitudes.