A satellite is in orbit about Earth. Its orbital radius is 7.51×107 m. The mass of the satellite is 5415 kg and the mass of Earth is 5.974×1024 kg. Determine the orbital speed of the satellite in mi/s. 1 mi/s = 1609 m/s.

The velocity required to maintain a circular orbit around the Earth may be computed from the following:

Vc = sqrt(µ/r)

where Vc is the circular orbital velocity in feet per second, µ (pronounced meuw as opposed to meow) is the gravitational constant of the Earth, ~3.986365x10^14 ft.^3/sec.^2, and r is the distance from the center of the earth to the altitude in question in feet.

Therefore,
V = sqrt[3.986365x10^14/7.51x10^7]

I am confused because the answer should be in mi/s.

My apologies for the typo.

The velocity required to maintain a circular orbit around the Earth may be computed from the following:

Vc = sqrt(µ/r)

where Vc is the circular orbital velocity in meters per second, µ (pronounced meuw as opposed to meow) is the gravitational constant of the Earth, ~3.986365x10^14 met.^3/sec.^2, and r is the distance from the center of the earth to the altitude in question in meters.

Therefore,
V = sqrt[3.986365x10^14/7.51x10^7]
= 2303.926 met./sec.x3.281ft./met.
= 7559 ft./sec.x1mile/5280ft.
= 1.4316miles/sec.x3600sec./1 hour
= 5154 mi/hr.

To determine the orbital speed of the satellite, we can use the formula for the orbital speed:

V = √(G * M / R)

Where:
V = orbital speed
G = gravitational constant (6.67430 × 10^-11 m^3⋅kg^-1⋅s^-2)
M = mass of the Earth
R = radius of the satellite's orbit

First, convert the orbital radius from meters to kilometers:
7.51 × 10^7 m = 7.51 × 10^7 / 1000 km = 7.51 × 10^4 km

Now, convert the orbital speed from km/s to mi/s:
1 km/s = 0.621371 mi/s

Plugging in the values:
V = √(6.67430 × 10^-11 * 5.974 × 10^24 / (7.51 × 10^4 * 10^3))

Simplifying:
V = √(4.0032395 × 10^14 / 7.51 × 10^7)

V = √(5.33055929 × 10^6) m/s

Now, convert the orbital speed from m/s to mi/s:
1 m/s = 0.000621371 mi/s

V = √(5.33055929 × 10^6 * 0.000621371) mi/s

V ≈ √(3309.38725188954) mi/s

V ≈ 57.4871705185 mi/s, rounded to the nearest hundredth.

Therefore, the orbital speed of the satellite is approximately 57.49 mi/s.