Two planets, X and Y , travel counterclockwise in circular orbits about a star, as shown in
figure below. The radii of their orbits are in the ratio 3 : 1. At some time, they are aligned as in the
figure, making a straight line with the star. During the next five years, the angular displacement of
planet X is �
2 , as shown in figure. Where is planet Y at this time?
3
To determine the position of planet Y at this time, we need to consider the angular displacement of planet X and the ratio of their radii.
Let's break down the problem into steps:
1. Start by understanding the given information:
- Planet X and Y travel counterclockwise in circular orbits around a star.
- The radii of their orbits are in the ratio 3:1.
- They are aligned as a straight line with the star at some time.
- The angular displacement of planet X during the next five years is 2/3.
2. Understand the concept of angular displacement:
Angular displacement measures the change in the angle as an object moves. A full circle consists of 360 degrees or 2π radians. So, an angular displacement of 2/3 indicates that planet X has moved 2/3 of a full circle.
3. Determine the position of planet Y:
Since planet X has moved 2/3 of a full circle, we need to find out where planet Y is in relation to planet X and the star.
- Start by understanding the ratio of their radii: 3:1.
- Since the radius ratio is 3:1, the circumference of planet Y's orbit is three times that of planet X's orbit, assuming both have the same angular displacement.
- Divide the 2/3 of a full circle by 3 to find the angular displacement of planet Y.
(2/3) / 3 = 2/9
Therefore, planet Y is at an angular displacement of 2/9 of a full circle from the straight line alignment.
4. Convert the angular displacement into a fraction of a full circle:
Since we have found that planet Y is at an angular displacement of 2/9, we can convert it to a fraction of a full circle by multiplying it by 360 degrees or 2π radians.
(2/9) * 360 = 80 degrees
(2/9) * 2π = (4π/9) radians
Therefore, planet Y is 80 degrees or (4π/9) radians away from the straight line alignment.
In summary, at the given time, planet Y is located at an angular displacement of 2/9 of a full circle from the straight line alignment with the star.