I don't know yet how to integrate this function....

[pabilbado]

I was just wondering about the path an orbiting particle takes when it's center of circular motion also translates. So as to calculate the distance traveled I know I have to get a function that relates velocity with respect to time and integrate it, but I can't doit, I get stuck integrating. I have search the Internet across and nothing. Possibly I don't know what I am looking for. Here is the function:

I can simplify it but it continues being difficult:

 

[Dr. Courtney]

Is this a homework problem?

 

[pabilbado]

Is this a homework problem?

No, is not. Two days ago it was my father's birthday (he is a nerdy engineer so he will like this kind of present) so I am trying to calculate how much kilometers he has traveled around the sun. And as an approximation I have made the Earth's orbit a straight line, and this is the result of a point on Earth rotating and translating.

 

[SteamKing]

No, is not. Two days ago it was my father's birthday (he is a nerdy engineer so he will like this kind of present) so I am trying to calculate how much kilometers he has traveled around the sun. And as an approximation I have made the Earth's orbit a straight line, and this is the result of a point on Earth rotating and translating.

I don't know how you can make an orbit a straight line.

If you want to know how far your father has traveled in his lifetime, assume that Earth's orbit is a circle with a radius of 150 million kilometers. Every year, the Earth makes one orbit of the sun, by definition. The eccentricity of Earth's orbit is about 0.017, so a circle is a good approximation for all practical purposes here.

 

[pabilbado]

I don't know how you can make an orbit a straight line.

If you want to know how far your father has traveled in his lifetime, assume that Earth's orbit is a circle with a radius of 150 million kilometers. Every year, the Earth makes one orbit of the sun, by definition. The eccentricity of Earth's orbit is about 0.017, so a circle is a good approximation for all practical purposes here.

Yeah, that's the easy way. What I want is to take into account the rotation of earth, while is moving around the sun. Because a point in the surface of the Earth does not travel a circular path around the sun but rather a really "curly" trajectory. So as a first approximation I have flattened Earth's orbit so it is just traveling in a straight line and I need to take into consideration the that the point is also rotating. That wwhat the integral is for.

 

[Dr. Courtney]

Why not write the parametric equations as:

x(t) = Ra*cos(2*pi t/Ta) + Rb*cos(2*pi*t/Tb)*cos(latitude)
y(t) = Ra*sin(2*pi t/Ta) + Rb*sin(2*pi*t/Tb)*cos(latitude)

Use the arc length integral, substitute in, and integrate numerically.

Ra, Ta = radius and period of Earth's orbit
Rb = Earth's radius, Tb = one day

Hopefully, you can treat the latitude as constant.

 

[pabilbado]

Why not write the parametric equations as:

x(t) = Ra*cos(2*pi t/Ta) + Rb*cos(2*pi*t/Tb)*cos(latitude)
y(t) = Ra*sin(2*pi t/Ta) + Rb*sin(2*pi*t/Tb)*cos(latitude)

Use the arc length integral, substitute in, and integrate numerically.

Ra, Ta = radius and period of Earth's orbit
Rb = Earth's radius, Tb = one day

Hopefully, you can treat the latitude as constant.

EDIT: thinking about it I believe I will be stuck with the same problem: integrating a function that has a trigonometric function inside a square root.

Yeah I thought about it, but I had two different directions, but I wasn't sure if I could just use the arc length for the first equation, then for the second one and sum them over. Is that what I have to do? If it wasn't the case just to satisfy my curiosity how could I integrate the first function I posted?(My approximate scenario would also yield a function for the movement of a particle around a translating and rotating sphere, wouldn't it?).

 

[pabilbado]

I forgot to post how I solved the problem. I did the taylor series and then I integrated term by term. I did it a long time ago, but i forgott to post it here.

 

[Fervent Freyja]

Something so thoughtful and sweet should have a place on the refrigerator forever!