Understanding Orbital Motion
To determine the ratios of time periods and speeds of two planets orbiting the sun, we can use Kepler's Third Law and the concept of circular motion.
1. Distance from the Sun
- Planet A: 10^13 m
- Planet B: 10^11 m
2. Time Period Ratio
According to Kepler's Third Law, the square of the time period (T) of a planet is directly proportional to the cube of its average distance (r) from the sun:
- T^2 ∝ r^3
Thus, the ratio of the time periods (T_A / T_B) can be expressed as:
- (T_A / T_B) = (r_A / r_B)^(3/2)
Calculating the distances:
- r_A = 10^13 m
- r_B = 10^11 m
Now substituting the values:
- (T_A / T_B) = ((10^13) / (10^11))^(3/2) = (10^2)^(3/2) = 10^3
Therefore, the ratio of time periods is:
- T_A : T_B = 1000 : 1
3. Speed Ratio
The orbital speed (v) of a planet is given by the formula:
- v = √(GM/r)
Where G is the gravitational constant and M is the solar mass, which are constant for both planets. The ratio of speeds (v_A / v_B) can be expressed as:
- (v_A / v_B) = √(r_B / r_A)
Calculating the ratio:
- (v_A / v_B) = √((10^11) / (10^13)) = √(1/100) = 1/10
Thus, the speed ratio is:
- v_A : v_B = 1 : 10
4. Summary
- Time Periods Ratio: 1000 : 1
- Speeds Ratio: 1 : 10
This analysis shows how significantly the distance affects both the time period and speed of the planets in their orbits.