How Does Earth's Orbit Affect Light Travel Time?

[ahrog]

Homework Statement

Ole Roemer found that the maximum increased delay in the appearance of Io (a moon of Jupiter, I believe) from one orbit to the next was 14 seconds.
a) How far does light travel in 14 seconds?
b) Each orbit of Io is 42.5 h. Earth traveled the distance calculated above in 42.5 h. Find the speed of Earth in km/s.
c) See if your answer for part b is reasonable. Calculate Earth's speed in orbit using the orbital radius, 1.5 x 10^8 km and the period, one year.

Homework Equations

v=d/t

The Attempt at a Solution

a) I got it to be 4.2 x 10^9 m and I'm pretty confident about that answer, so that isn't where I need help.
b) If Earth traveled the same distance as in a) in 42.5 h, I'm assuming I just go V=d/t where the answer is 27451 m/s...
c) For this, I just can't think of the steps. What formula for orbital motion do I need?

 

[mplayer]

Your answers in (A) and (B) look right to me.

C) I think they want you to assume the Earth is moving along a perfectly circular orbit around the sun (its really slightly elliptical, but I think you can ignore that for this question). Using the given radius (distance from Earth to sun), you can calculate the circumference of that circle (orbit). This is the total distance the Earth travels over the given period of a year. You'll need to convert year into its equivalent amount of seconds.

Now you can use: v = d / t

Hint: watch your units

I got an answer that is comparable to the answer in (B) so hopefully that means its right heh.

Hope that helps :)

 

[nlightner]

a) Your calculation for the distance light travels in 14 seconds is correct. It is important to note that this value is an approximation and can vary slightly depending on the speed of light at the time of measurement.

b) To calculate the speed of Earth, we can use the formula v = d/t. Since we know the distance traveled in 42.5 hours (4.2 x 10^9 m) and the time it took (42.5 h), we can plug these values in to get v = (4.2 x 10^9 m)/(42.5 h) = 9.88 x 10^7 m/s. To convert this to km/s, we divide by 1000, giving us a final answer of 98.8 km/s.

c) To calculate Earth's speed in orbit using the given orbital radius and period, we can use the formula v = 2πr/T, where r is the orbital radius and T is the period. Plugging in the values, we get v = 2π(1.5 x 10^8 km)/(1 year) = 2.98 x 10^4 km/s. This is a much larger value than the one calculated in part b, which is expected since the orbital speed of Earth is much faster than its speed of rotation. This confirms that our answer in part b is reasonable.