A satellite moves in a circular orbit around
the Earth at a speed of 5.9 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 · m^2
/kg^2
.
Answer in units of km
To determine the satellite's altitude above the surface of the Earth, we can use the concept of centripetal force and gravitational force.
The centripetal force acting on the satellite is provided by the gravitational force between the satellite and the Earth. The centripetal force is given by the equation:
F_c = m * v^2 / r
where F_c 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 satellite's orbit (which is the sum of the Earth's radius and the altitude of the satellite).
The gravitational force between the satellite and the Earth is given by the equation:
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 satellite and the center of the Earth.
Setting the centripetal force equal to the gravitational force, we have:
m * v^2 / r = G * (m * M) / r^2
Simplifying the equation, we get:
v^2 = G * M / r
Rearranging the equation to solve for r, we have:
r = G * M / v^2
Substituting the given values and performing the calculations:
r = (6.67259 × 10^-11 N · m^2/kg^2 * 5.98 × 10^24 kg) / (5.9 km/s)^2
r = (39.82 × 10^13 N · m^2/kg) / (34.81 × 10^6 m^2/s^2)
r = 1.143 × 10^7 m
Finally, to convert the distance to kilometers (km):
altitude = r - 6370 km
altitude = 1.143 × 10^7 m - 6370 km
altitude = 11,000 km (rounded to the nearest kilometer)
Therefore, the satellite's altitude above the surface of the Earth is approximately 11,000 km.
To determine the satellite's altitude above the surface of the Earth, we need to use the concept of centripetal force and gravitational force.
First, we calculate the gravitational force between the satellite and the Earth using the formula:
F_gravity = (G * m1 * m2) / r^2
where G is the universal gravitational constant, m1, and m2 are the masses of the two objects (Earth and satellite), and r is the distance between their centers.
Next, we equate the gravitational force to the centripetal force, as the satellite is moving in a circular orbit. The centripetal force is given by:
F_centripetal = (m * v^2) / r
where m is the mass of the satellite, v is the velocity, and r is the radius of the circular orbit.
Setting these two forces equal, we have:
(G * m1 * m2) / r^2 = (m * v^2) / r
Rearranging the equation to solve for r, we get:
r = (G * m1 * m2) / (v^2)
Now, substituting the given values:
G = 6.67259 × 10^(-11) N · m^2/kg^2
m1 (mass of the Earth) = 5.98 × 10^24 kg
m2 (mass of the satellite) = m (we don't have this information)
v (velocity) = 5.9 km/s = 5900 m/s
r = [(6.67259 × 10^(-11) N · m^2/kg^2) * (5.98 × 10^24 kg)] / (5900 m/s)^2
Solving this equation will give us the radius of the orbit, which is the sum of the radius of the Earth and the altitude of the satellite above the surface of the Earth.
Finally, to find the satellite's altitude, subtract the radius of the Earth (6370 km) from the calculated radius of the orbit.
Note: Since we don't have the mass of the satellite, we won't be able to calculate the exact altitude. However, the radius of the orbit will give us the distance from the center of the Earth to the satellite.
Let me know if you would like me to calculate the radius of the orbit with the given values.