Calculate the speed of a satellite moving in a stable
circular orbit about the Earth at a height of 4800 km.
To calculate the speed of a satellite in a stable circular orbit around the Earth, we can use the concept of centripetal force.
1. First, we need to find the gravitational force acting on the satellite at that height. We can use the formula for gravitational force:
F = (G * m * M) / r^2
Where:
F is the gravitational force
G is the gravitational constant (6.67430 * 10^-11 N m^2/kg^2)
m is the mass of the satellite
M is the mass of the Earth
r is the distance between the center of the Earth and the satellite (radius of Earth + height of satellite)
We can assume the mass of the satellite is negligible compared to the mass of the Earth.
2. The gravitational force acting on the satellite provides the centripetal force required for circular motion. Therefore, it is equal to the formula for centripetal force:
F = m * v^2 / r
Where:
F is the gravitational force (from step 1)
m is the mass of the satellite (negligible for this calculation)
v is the velocity or speed of the satellite
r is the distance between the center of the Earth and the satellite (radius of Earth + height of satellite)
We can equate these two equations and solve for v:
(G * m * M) / r^2 = m * v^2 / r
Since the satellite mass is negligible and cancels out on both sides, we can simplify the equation to solve for v:
v^2 = (G * M) / r
v = sqrt((G * M) / r)
3. Now we can substitute the values into the equation. The mass of the Earth (M) is 5.972 × 10^24 kg, the radius of the Earth (r) is 6,378 km, and the height of the satellite (h) is 4800 km.
v = sqrt((6.67430 * 10^-11 N m^2/kg^2 * 5.972 × 10^24 kg) / (6378000 m + 4800000 m))
Convert the radius of the Earth and height of the satellite to meters:
v = sqrt((6.67430 * 10^-11 N m^2/kg^2 * 5.972 × 10^24 kg) / (11178000 m))
Calculate:
v = sqrt((3.9877 * 10^14 N m^2/kg) / (11178000 m))
Simplify:
v = sqrt(3.5631 * 10^4 m^2/s^2)
v ≈ 1886 m/s
Therefore, the speed of the satellite moving in a stable circular orbit about the Earth at a height of 4800 km is approximately 1886 m/s.
Sure! To calculate the speed of a satellite in a stable circular orbit around the Earth, we can use the formula for the speed of an object in circular motion:
v = √(GMe / r)
where:
v is the velocity of the satellite,
G is the gravitational constant (approximately 6.674 × 10^-11 m^3 kg^-1 s^-2),
Me is the mass of the Earth (approximately 5.972 × 10^24 kg), and
r is the radius of the satellite's orbit.
In this case, the satellite orbits at a height of 4800 km above the Earth's surface. To find the radius of the orbit (r), we need to add the radius of the Earth to the height of the satellite.
The radius of the Earth is approximately 6371 km, so the radius of the orbit can be calculated as follows:
r = radius of the Earth + height of the satellite
r = 6371 km + 4800 km
r = 11171 km
Converting this radius to meters:
r = 11,171 km * 1000 m/km
r = 11,171,000 m
Now we can substitute the values into the formula to find the speed:
v = √(GMe / r)
v = √((6.674 × 10^-11 m^3 kg^-1 s^-2) * (5.972 × 10^24 kg) / (11,171,000 m))
Calculating this expression:
v ≈ √((4.002504 × 10^14 m^3 kg^-1 s^-2) / 11,171,000 m)
v ≈ √(3.579 × 10^4 m^2 s^-2)
v ≈ 189.4 m/s
Therefore, the speed of the satellite moving in a stable circular orbit about the Earth at a height of 4800 km is approximately 189.4 m/s.