A satellite moves in a circular orbit around the Earth at a speed of 5.8 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/
the gravitational force is the centripetal force
When orbiting at 5.8km/s:
From Vc = sqrt(µ/r) where Vc = the velocity of an orbiting body, µ = the gravitational constant of the earth and r the radius of the circular orbit,with µ = GM, G = the universal gravitational constant and M = the mass of the central body, the earth in this instant,
r = µ/Vc^2
= 6.67259x10^-11(5.98x10^24)/5800^2
The altitude is therefore
(r - 6370)/1000 km.
To determine the satellite's altitude above the surface of the Earth, we can use the relationship between the gravitational force and the centripetal force.
1. First, let's calculate the centripetal force acting on the satellite. The centripetal force is given by the formula:
Fc = m * v^2 / r
where Fc is the centripetal force, m is the mass of the satellite, v is the velocity of the satellite, and r is the radius of the circular orbit.
2. We know the speed of the satellite is 5.8 km/s. We need to convert it to m/s. There are 1000 meters in a kilometer, so:
v = 5.8 km/s * 1000 m/km = 5800 m/s
3. The mass of the satellite does not affect its altitude, so we can ignore it for this calculation.
4. Now, let's calculate the gravitational force acting on the satellite. The gravitational force is given by the formula:
Fg = G * m * M / r^2
where Fg 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.
5. The mass of the Earth is given as 5.98 × 10^24 kg.
6. Rearranging the gravitational force formula, we can solve for r:
r = sqrt(G * M / (Fc / m))
7. Substituting the values into the formula:
r = sqrt((6.67259 × 10^-11 N · m^2/kg^2) * (5.98 × 10^24 kg) / (Fc / m))
8. Plugging in the centripetal force value:
r = sqrt((6.67259 × 10^-11 N · m^2/kg^2) * (5.98 × 10^24 kg) / ((m * v^2) / r))
9. Simplifying the equation:
r = sqrt((6.67259 × 10^-11 N · m^2/kg^2) * (5.98 × 10^24 kg) / v^2)
10. Plugging in the known values:
r = sqrt((6.67259 × 10^-11 N · m^2/kg^2) * (5.98 × 10^24 kg) / (5800 m/s)^2)
11. Solving the equation:
r ≈ sqrt(3.98628 × 10^14 m^3/s^2 / 33640000 m^2/s^2)
r ≈ sqrt(1186.61)
r ≈ 34.47 x 10^6 m
So, the satellite's altitude above the surface of the Earth is approximately 34.47 x 10^6 meters or 34,470 kilometers.
To determine the satellite's altitude above the surface of the Earth, we first need to find the centripetal force acting on the satellite.
The centripetal force is provided by the gravitational force between the satellite and the Earth. We can equate the gravitational force with the centripetal force to find the relationship between them.
The gravitational force between two objects can be calculated using the equation:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the universal gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between the centers of the objects.
In this case, the satellite is moving in a circular orbit around the Earth, so the gravitational force provides the necessary centripetal force.
The centripetal force can be calculated using the equation:
Fc = m * v^2 / r
where Fc is the centripetal force, m is the mass of the satellite, v is its speed, and r is the radius of the circular orbit (which is the sum of the Earth's radius and the satellite's altitude).
Setting the gravitational force equal to the centripetal force, we have:
(G * m * Msatellite) / r^2 = m * v^2 / r
The mass of the satellite (Msatellite) cancels out, so we are left with:
(G * me) / r^2 = v^2 / r
Simplifying further, we get:
r^3 = (G * me) / v^2
To find the satellite's altitude, we need to solve for r. Rearranging the equation, we have:
r = [(G * me) / v^2]^(1/3)
Now we can substitute the given values into the equation:
G = 6.67259 × 10^-11 N · m^2/kg^2
me = 5.98 × 10^24 kg
v = 5.8 km/s = 5800 m/s
Plugging these values into the equation and solving for r, we get:
r = [((6.67259 × 10^-11 N · m^2/kg^2) * (5.98 × 10^24 kg)) / (5800 m/s)^2]^(1/3)
Calculating this expression will give us the satellite's altitude above the surface of the Earth.