Circular Motion with Angular Speed dealing with planets

[Garrant3]

1. Homework Statement
Mars orbits the sun at a mean distance of 228 million km, in a period of 687 days. The Earth orbits at a mean distance of 149.6 million km in a period of 365.26 days. All answers should be in the range (0, 2pi)
a) suppose Earth and Mars are positioned such that Earth lies on a straight line between Mars and the Sun. Exactly 365.26 days later, when the Earth has completed one orbit, what is the angle between the Earth-Sun line and the Earth-Mars line?
b) The initial situation in part a) is a closest approach of Mars to the Earth. What is the time between 2 closest approaches? Assume constant orbital speeds and circular orbits. (Hint: when angles are equal)
c) Another way of expressing b) is with the angle between the Sun, Earth, and Mars in the two closest approach situations. What is that angle?


2. Homework Equations
Probably angular speed= 2pi/period
angle traveled = angular speed x time



3. The Attempt at a Solution
My homework is online, and we get three tries per problem. I attempted this one and apparently got all three parts wrong. For a) I used angular speed = 2pi/period to get the angular speeds for Earth and Mars. I then multiplied these speeds by time to get the angle traveled for each. Earth was 2pi, and for Mars I got 3.34 radians, which is wrong. :/
For part b) It says to use the relationship between angle, angular speed,and time, which is angle traveled = angular speed x time. The angle traveled by Earth would be angle traveled by Mars + 2pi according to the hint.
c) According to the hint. solve the equation used in b) for the angle instead of for time. I think once I get
I figure once I get a) I may be able to eventually work out the rest. Anyone know what I did wrong with a)? I've tried so many different ways but keep getting the same answer.
Thanks!

 

[tiny-tim]

Welcome to PF!

Hi Garrant3! Welcome to PF!

(have a pi: π )

erm … read the question! …

… what is the angle between the Earth-Sun line and the Earth-Mars line?

 

[Garrant3]

Hi! Thanks for the welcome!
I'm sorry, it is supposed to be "between the Earth-Sun line and the Mars-Sun line" instead of "between the Earth-Sun line and the Earth-Mars line".
I'm still working on it too, so let me know what you think. Maybe instead I should take 2pi - 3.34 to get the angle?
Once again, thanks for responding! :)

 

[Garrant3]

Sorry for double posting, but I just figured it out!
If anyone is interested, it did work taking 2pi - 3.34.

 

[rwisz]

I would like to clarify a few things about the concept of circular motion with angular speed when dealing with planets. Firstly, the angular speed of a planet is not constant throughout its orbit. It varies depending on its distance from the sun, as described by Kepler's Second Law. Therefore, using the formula angular speed = 2pi/period may not give an accurate answer.

To solve part a) of this problem, we can use the concept of angular velocity, which is the rate of change of angular displacement. In this case, the Earth and Mars are moving in the same direction, so the angle between the Earth-Sun line and the Earth-Mars line will not change. Therefore, the angle between these two lines after 365.26 days will be the same as the angle between these two lines at the initial position. This angle can be calculated by finding the difference between the angular displacement of Earth and Mars after 365.26 days.

For part b) and c), the closest approach between Mars and Earth does not necessarily happen after exactly one orbit. It depends on the initial positions of the planets. Also, the angular speed of the planets will not be constant during the time between two closest approaches. Therefore, using the formula angle traveled = angular speed x time may not give an accurate answer. Instead, we can use Kepler's Third Law, which states that the ratio of the squares of the orbital periods of two planets is equal to the ratio of the cubes of their semi-major axes. Using this law, we can find the time between two closest approaches.

I hope this helps clarify the concept of circular motion with angular speed when dealing with planets. It is important to consider the varying angular speed and non-constant orbits of planets when solving such problems.