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 Mars-Sun line? (in rad)
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 for both Mars and Earth. (Hint: when angles are equal) (in days)
c) Another way of expressing the answer to part (b) is in terms of the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations. What is that angle? (in rad)
Physics (please help!!!!) - drwls, Friday, April 8, 2011 at 2:12am
a) When Earth has travelled one orbit, Mars will have completed 365.2/687 = 53.2% of an orbit. Mars will be 46.8% of an orbit behinbd Earth. Convert that to radians.
b) Extrapolate to find out how long it takes for Mars to fall 360 degrees behond Earth. That is when a new alignment will occur.
c) Figure out how far (in angle) Earth travels between alignments. It will be more than two complete revolutions. You only want the additional angle beyond two revolutions. (Use the time from part (b)). Convert that to radians.
For part A I got 2.94 rad, not sure if that's right... I still don't get part b... I just used the initial distance I got for Mars after 365.25 days and then multiplyed it by 3 and I got 363 days... I don't think that makes sense... I can't solve part c without part b.
Ok I got the 2 first ones right... 2.94 and my new value for b is 777.... I still cant get the last one, I thought that by multiplying Mars' angular speed and the time I got for part A that was going to be the answer.. I got 7.11 rad but that's wrong....
If I substract 2pi from 7.11 I get 0.83... will that be the angle???
But 7.11 is just more thatn one complete revolution, then that's not the right anwer?
Oh wait, I was using the angular speed of Mars, using the angular speed of Earth I get 13.4 radians and that will be more than two revolutions so the additional angle is 0.798 rad? Is that right?? Please someone answer...
a) To find the angle between the Earth-Sun line and the Mars-Sun line, you need to calculate the difference in their angular positions after 365.26 days.
Since Mars takes 687 days to complete one orbit, it would have completed (365.26 / 687) * 2π radians of its orbit. This gives us the proportion of one complete orbit that Mars has completed after 365.26 days.
To find the angle, we can multiply this proportion by 2π radians since a full orbit is 2π radians.
Angle = (365.26 / 687) * 2π radians = 1.891 radians (approximately)
So the angle between the Earth-Sun line and the Mars-Sun line after 365.26 days is approximately 1.891 radians.
b) To find the time between two closest approaches, we need to determine how long it takes for Mars to fall 360 degrees behind Earth.
Since Earth takes 365.26 days to complete one orbit, it would have completed (365.26 / 687) * 360 degrees of its orbit after the time it takes for Mars to fall 360 degrees behind.
To find the time, we can multiply this proportion by the period of Earth's orbit, which is 365.26 days.
Time = (365.26 / 687) * 365.26 days = 194.95 days (approximately)
So the time between two closest approaches is approximately 194.95 days.
c) To find the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations, we need to find out how far Earth travels between the alignments.
Since Earth takes 365.26 days to complete one orbit, it would have traveled 360 degrees in that time.
To find the additional angle beyond two complete revolutions, we subtract 360 degrees from the total angle traveled by Earth after the time it takes for Mars to fall 360 degrees behind.
Angle = (365.26 / 687) * 360 degrees - 360 degrees
Angle = 186.7 degrees (approximately)
To convert this angle to radians, we multiply by (π / 180).
Angle = 186.7 * (π / 180) radians
So the angle between the line drawn through the Sun, Earth, and Mars in the two closest approach situations is approximately 3.258 radians.