Scientists want to place a 3100 kg satellite in orbit around Mars. They plan to have the satellite orbit a distance equal to 1.9 times the radius of Mars above the surface of the planet. Here is some information that will help solve this problem:
mmars = 6.4191 x 1023 kg
rmars = 3.397 x 106 m
G = 6.67428 x 10-11 N-m2/kg2
1)What should the radius of the orbit be (measured from the center of Mars), if we want the satellite to take 8 times longer to complete one full revolution of its orbit?
so far I have found speed to orbit 2085m/s and 8.2hrs to complete one revolution
I have no clue how to solve this or what to plug in where
Ac = omega^2 R
m Ac = G m Mmars /R^2
so
omega^2 R = G Mmars/R^2
omega^2 = G Mmars/R^3
a circle is when omega T = 2 pi
or
T ^2 = 4 pi^2/omega^2 = 4 pi^2 R^3 /G Mmars
Tbig^2 = 64 T^2
so
Rbig^3 = 64 R^3
Rbig/R = 64^(1/3) or 4 times the radius
To solve this problem, we can use the principle of conservation of angular momentum.
Angular momentum (L) is given by the formula L = mvr, where m is the mass of the satellite, v is its velocity, and r is the radius of the orbit.
Since the satellite takes 8 times longer to complete one full revolution of its orbit, its period (T) will be 8 times longer than the period of a regular orbit.
The period of a regular orbit (T0) can be calculated using the formula T0 = 2πr/v, where π is a mathematical constant (~3.14159).
Let's start by finding the velocity of the satellite. You mentioned that you have already found the speed, which is 2085 m/s. To find the velocity (v), we need to divide the speed by the square root of 2, as the speed is the magnitude of velocity in an elliptical orbit. So, v = 2085 m/s divided by the square root of 2.
Now, we can use the period T and the radius of the orbit r to solve for the angular momentum L.
The angular momentum of the satellite in the regular orbit (L0) can be calculated using the formula L0 = mvr.
Since we want the satellite to take 8 times longer to complete one full revolution, the angular momentum of the satellite in the new orbit (L) will equal L0 divided by 8. L = L0/8.
Since angular momentum is conserved, L = mvr = L0/8.
Now, we need to solve for the radius of the new orbit (r). Rearranging the formula, we get r = L/(mv).
Substituting L0/8 in place of L, and the values you have (mass of Mars, mass of the satellite, velocity), we can plug them into the formula to solve for the radius (r).