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

In summary, the engineer was trying to find out how far his father had traveled in his lifetime by assuming an orbit that approximated a circle. He calculated that the Earth makes one orbit of the sun every year and that the eccentricity of Earth's orbit is about 0.017. He also assumed that the latitude of the point on Earth being rotated and translated stayed the same. Finally, he solved for x and y using the arc length integral and integrated numerically.
  • #1
pabilbado
Gold Member
17
1
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:
df40425500452af6d860f2e1c2e4efdb.png

I can simplify it but it continues being difficult:
ba863ee3701d465bae822f56ed19c330.png
 
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  • #3
Dr. Courtney said:
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.
 
  • #4
pabilbado said:
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.
 
  • #5
SteamKing said:
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.
 
  • #6
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.
 
  • #7
Dr. Courtney said:
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?).
 
  • #8
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.
 
  • #9
Something so thoughtful and sweet should have a place on the refrigerator forever! :smile:
 

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

What is the process for integrating a function?

The process for integrating a function involves finding the antiderivative of the function, also known as the indefinite integral. This can be done using various techniques such as substitution, integration by parts, or using specific integration rules.

Why is integrating a function important?

Integrating a function allows us to find the total value or area under a curve, which can have many practical applications in fields such as physics, engineering, and economics. It is also a fundamental concept in calculus and helps us understand the relationship between a function and its derivative.

How do I know which integration technique to use?

Choosing the right integration technique can depend on the form of the function and the specific problem at hand. It is important to have a good understanding of the different integration techniques and their applications. Practice and familiarity with various types of functions can also help in determining the appropriate technique.

What if I can't find the antiderivative of a function?

There are some functions that do not have an antiderivative in terms of elementary functions. In such cases, we can use numerical methods or approximation techniques to find an approximate value for the integral. These methods involve dividing the area under the curve into smaller sections and calculating the sum of these sections to get an estimate of the integral.

Can I use a calculator to integrate a function?

Yes, there are many calculators and computer programs that can perform integration for you. However, it is important to have a good understanding of the concepts and techniques involved in integration in order to use these tools effectively and interpret the results correctly.

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