The satellite is in a circular orbit 400 km above the Earth's surface, where the free-fall acceleration is 8.71 m/s2. The radius of the Earth is 6400 km. Determine the speed of the satellite.

mv²/(R+h)=g

v=sqrt{g(R+h)/m}

7580m/s^2

To determine the speed of the satellite, we can use the formula for the centripetal acceleration of an object in circular motion:

a = v^2 / r

Where:
a = centripetal acceleration
v = velocity (speed) of the satellite
r = radius of the circular orbit

In this case, the centripetal acceleration is equal to the free-fall acceleration (g), which is 8.71 m/s^2. The radius (r) is the sum of the altitude of the satellite (400 km) and the radius of the Earth (6400 km).

So, let's calculate the speed of the satellite:

First, convert the altitude and Earth's radius to meters:
Altitude = 400 km = 400,000 m
Radius of Earth = 6400 km = 6,400,000 m

Next, add the altitude and radius to get the radius of the circular orbit:
r = altitude + radius of Earth
r = 400,000 m + 6,400,000 m
r = 6,800,000 m

Now, rearrange the formula to solve for velocity (v):
v = √(a * r)

Substitute the values into the formula:
v = √(8.71 m/s^2 * 6,800,000 m)

Now, calculate the velocity using a calculator or by solving the equation step by step:
v ≈ 7,917.5 m/s

Therefore, the speed of the satellite is approximately 7,917.5 m/s.