Let G be the universal gravitational constant and mp be the mass of the planet a satellite is orbiting. Which equation could be used to find the velocity of the satellite if it is placed in a geostationary orbit?(1 point)
Responses
v=Gmp/(7,324 km)−−−−−−−√
v=Gmp/(15,522 km)−−−−−−−−√
v=Gmp/(48,115 km)−−−−−−−−√
v=Gmp/(42,164 km)−−−−−−−−√
v = Gmp/(42,164 km)
The correct equation to find the velocity of a satellite in a geostationary orbit is:
v = Gmp / (42,164 km)^(1/2)
To find the equation that could be used to find the velocity of the satellite in a geostationary orbit, we need to understand the concept of geostationary orbit and the equation that relates it to the given variables.
A geostationary orbit is an orbit around the planet where a satellite appears to be stationary relative to the Earth's surface. In this orbit, the satellite's orbital period matches Earth's rotation period, resulting in the satellite remaining fixed above a certain location on Earth's equator.
The equation that relates the velocity of a satellite in a circular orbit to the mass of the planet it is orbiting can be derived from the centripetal force. The centripetal force required to keep the satellite in orbit is the gravitational force between the satellite and the planet.
This can be expressed as:
F_gravity = F_centripetal
G * (mp * ms) / r^2 = ms * v^2 / r
Where:
G is the universal gravitational constant,
mp is the mass of the planet,
ms is the mass of the satellite (usually negligible compared to the planet's mass),
r is the radius of the orbit (distance from the center of the planet to the satellite),
and v is the velocity of the satellite.
Simplifying the equation, we can solve for v:
v^2 = G * mp / r
v = √(G * mp / r)
Now, we can compare this equation to the given options:
v = G * mp / (7,324 km)^(1/2)
v = G * mp / (15,522 km)^(1/2)
v = G * mp / (48,115 km)^(1/2)
v = G * mp / (42,164 km)^(1/2)
Comparing these options to the derived equation, we can see that the correct equation to find the velocity of the satellite in a geostationary orbit is:
v = G * mp / (42,164 km)^(1/2)
Therefore, the correct option is "v = Gmp/(42,164 km)−−−−−−−√".