HELP. I need to understand this for a test tomorrow. THANKS
Consider planet mass of 4.98 E 21 kilograms with a satellite orbitting it at distance of 3.11 E 6 meters from planet's center. Satellite has mass of 369 kilograms.
How long (inseconds) does it take the satellite to make one complete orbit around planet.
I just can't seem to get the correct answer - Please show me step by step.
Thanks
Sure! Let's walk through the steps to find the answer.
To determine the time it takes for the satellite to complete one orbit around the planet, we can use Kepler's third law of planetary motion, which states that the square of the orbital period of a planet is directly proportional to the cube of the semimajor axis of its orbit.
Step 1: Find the gravitational constant (G).
The gravitational constant is approximately 6.67430 x 10^-11 m^3/kg/s^2.
Step 2: Find the total mass (M) of the planet and the satellite.
The total mass (M) is the sum of the mass of the planet and the mass of the satellite. In this case, M = mass of the planet + mass of the satellite.
M = 4.98 x 10^21 kg + 369 kg.
Step 3: Convert the distance from the planet's center to the semimajor axis (a).
The semimajor axis (a) is half of the distance between the closest and farthest points of the satellite's orbit around the planet. In this case, the distance provided is already the semimajor axis.
a = 3.11 x 10^6 meters.
Step 4: Use Kepler's third law to find the orbital period (T).
According to Kepler's third law, T^2 = (4π^2 / GM) * a^3.
Substitute the known values into the formula:
T^2 = (4π^2 / (6.67430 x 10^-11)) * (4.3980989353 x 10^18)^3.
Step 5: Solve for T (orbital period).
Take the square root of both sides of the equation to find the orbital period:
T = √((4π^2 / (6.67430 x 10^-11)) * (4.3980989353 x 10^18)^3).
Evaluating this expression will give you the orbital period of the satellite in seconds.
Remember to use your calculator to perform these calculations. Good luck with your test!