A satellite moves in a circular orbit around

the Earth at a speed of 5.3 km/s.
Determine the satellite’s altitude above
the surface of the Earth. Assume the
Earth is a homogeneous sphere of radius
6370 km and mass 5.98 × 1024 kg. The
value of the universal gravitational constant
is 6.67259 × 10−11 N · m2
/kg2
.
Answer in units of km

you know that

v^2 = GM^2 / (M+m)r

You have (or can easily calculate) G,M,m and r.

Plug and chug. Note that for m<<M, the approximate relation is

v^2 = GM/r

To determine the satellite's altitude above the surface of the Earth, we can use the centripetal force equation and the equation for the gravitational force.

The centripetal force acting on the satellite in its circular orbit is provided by the gravitational force between the Earth and the satellite.

The magnitude of the centripetal force is given by the equation:

F_c = m * a_c

where F_c is the centripetal force, m is the mass of the satellite, and a_c is the centripetal acceleration.

The gravitational force between the Earth and the satellite is given by the formula:

F_g = (G * m * M) / r^2

where F_g is the gravitational force, G is the universal gravitational constant, m is the mass of the satellite, M is the mass of the Earth, and r is the distance between the center of the Earth and the satellite.

Since the centripetal force is provided by the gravitational force, we can equate the two:

m * a_c = (G * m * M) / r^2

The mass of the satellite cancels out, so we have:

a_c = (G * M) / r^2

The centripetal acceleration can be expressed as:

a_c = v^2 / r

where v is the orbital velocity of the satellite.

Setting these two equations equal to each other:

(G * M) / r^2 = v^2 / r

Simplifying the equation:

r = (G * M) / v^2

Now we can substitute the given values into the equation:

G = 6.67259 × 10^(-11) N · m² / kg²
M = 5.98 × 10^24 kg
v = 5.3 km/s = 5.3 × 10^3 m/s

Plugging in the values:

r = (6.67259 × 10^(-11) N · m² / kg² * 5.98 × 10^24 kg) / (5.3 × 10^3 m/s)^2

Calculating the result:

r = 6.38 × 10^6 meters

Converting the result to kilometers:

r = 6.38 × 10^3 kilometers

Therefore, the satellite's altitude above the surface of the Earth is approximately 6.38 × 10^3 kilometers.