Final answer:
To find the new orbital period of a satellite at 1500 miles above Earth with a velocity of 38,000 mph, we use a proportional relationship comparing it to a known orbit. We set up the ratio using the direct variation with the orbit's radius and inverse variation with the orbital velocity, then solve for the unknown orbital period.
Step-by-step explanation:
The duration of a satellite's orbit around Earth can be described as varying directly with the orbit's radius and inversely with the orbital velocity. To calculate the orbital period for a new satellite trajectory, we need to compare it with a known situation using a proportional relationship. Given that a satellite orbits 660 miles (or 1062 kilometers) above Earth in 13 hours at 32,000 mph, we can set up a ratio to determine the orbital period at 1500 miles (or 2414 kilometers) above Earth with a velocity of 38,000 mph.
The ratio for the original orbit is T1 / (r1 / v1) = T2 / (r2 / v2), where T is time, r is radius (distance from Earth's center to the satellite), and v is velocity. Using the provided values, we get:
13 / ((3963 + 660) / 32000) = T2 / ((3963 + 1500) / 38000)
Solving for T2 gives us the new orbital period. Considering that we should convert the altitude to the same units and use Earth's radius in miles (3963 miles), the final calculation will provide us with the new orbital period in hours.