physica tow satellites are in orbit around a planet . satalite s1 takes 20 days to orbit the planet at a distance of 2x10 to the power of 5 km from the center of the planet . satalite s2 takes 160 days to orbit the planet
and the question is?
What is the distance of Satellite S2 from
the center of the planet?
To determine the required information about the planet and the satellites, we need to use Kepler's third law of planetary motion, which states that the square of the orbital period (T) of a planet or satellite is directly proportional to the cube of its average distance from the center of the planet (r^3).
Let's use this law to find the unknowns step by step:
1. Determine the orbital period (T1) and average distance (r1) of satellite S1:
- T1 = 20 days
- r1 = 2 × 10^5 km
2. Determine the orbital period (T2) of satellite S2:
- T2 = 160 days
Now, we can set up the equations using Kepler's third law:
For satellite S1:
T1^2 = k × r1^3
For satellite S2:
T2^2 = k × r2^3
Here, "k" is the constant of proportionality. Since both satellites orbit the same planet, the value of "k" will be the same for both equations.
3. Calculate the constant of proportionality (k):
- Plug the values of T1 and r1 into the equation for S1:
(20 days)^2 = k × (2 × 10^5 km)^3
400 days^2 = k × 8 × 10^15 km^3
- Rearrange the equation and solve for "k":
k = (400 days^2) / (8 × 10^15 km^3)
4. Determine the average distance (r2) of satellite S2:
- Plug the values of T2 and k into the equation for S2:
(160 days)^2 = k × r2^3
- Rearrange the equation and solve for "r2":
r2^3 = [(160 days)^2] / k
r2 = cube root of {[(160 days)^2] / k }
5. Calculate the average distance (r2):
- Substitute the calculated value of "k" and solve for "r2":
r2 = cube root of {[(160 days)^2] / [(400 days^2) / (8 × 10^15 km^3)]}
- Simplify the equation and calculate "r2":
r2 = 8 × 10^5 km
Therefore, satellite S2 has an average distance of 8 × 10^5 km from the center of the planet.