Definition of 'average inclination'?

  • Thread starter dilloncyh
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    Definition
In summary, the conversation discusses the concept of average inclination and how to calculate it for a given equation. Different methods of averaging, such as over horizontal distance or path distance, are mentioned and it is noted that they may produce different results. The conversation also includes a formula for calculating the average height of a hill and the use of the slope (gradient) function.
  • #1
dilloncyh
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It's not a homework problem, just a problem that suddenly popped out of my mind.


Homework Statement



So, my question is : how to calculate, or how to define 'average inclination'? Suppose I am given an equation y=f(x) that resembles the shape of a section of a hill, and I want to calculate the average inclination (something I always see when I watch cycling or alpine skiing race), how do I do that? Let's use f(x) = x^2 as an example for the following discussion.


Homework Equations



I know the average of a function is defined as:
92cdaca353a4fc0f69572d7770fbe734.png

So I suppose the correct equation should be similar, with f(x) the function that gives the shape of the hill I am calculating?

The Attempt at a Solution



Here comes the problem: Do I use dx for ds? And for (b-a) in the image, I need to replace it with the actual length of the function from x=a to x=b, right?
Since I don't really know I want to calculate (to find the slope at each point of the function and add them together, and then divide it by the total length of the slope?), my question may seem very silly, but please give me some idea.

thanks
 
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  • #2
The average slope is usually just the rise-over-run for two positions on the slope.
$$\frac{f(x+\Delta x)-f(x)}{\Delta x}$$ ... that would be what they did on the ski races.

You want to be a bit more detailed than that:

If ##y(x)## is the height of the slope at position ##x##,
Then $$\bar y = \frac{1}{b-a}\int_a^b y(x)\;\text{d}x$$ would be the average height of the hill in a<x<b.

The slope (gradient) function would be g(x)=dy/dx ... tells you the gradient at position x.
That help you think about it?
 
  • #3
I'm still a bit confused.

average slope = detla y / delta x seems very legit, but take y=x^2 and y=x as example.

If I want to calculate the average slope between x=0 and x=1, then by calculating the difference in height and horizontal displacement of the two end points, then both should give the same result (slope=1), but the length of the curve of y=x^2 is obviously longer than y=x (which is just sq root of 2), so by calculating the the sine of the 'triangle' if I straighten the curve section of y=x^2, I will get difference result.
 
  • #4
dilloncyh said:
I'm still a bit confused.

average slope = detla y / delta x seems very legit, but take y=x^2 and y=x as example.

If I want to calculate the average slope between x=0 and x=1, then by calculating the difference in height and horizontal displacement of the two end points, then both should give the same result (slope=1), but the length of the curve of y=x^2 is obviously longer than y=x (which is just sq root of 2), so by calculating the the sine of the 'triangle' if I straighten the curve section of y=x^2, I will get difference result.
It's a matter of how you choose to define the average. The usual would be to define it as average over horizontal distance, but as you note you could instead chose to define it as average over path distance.
 
  • #5
I'm with haruspex - there is no reason that two different averaging methods should produce the same value.
They are just producing a different kind of average.
 

Related to Definition of 'average inclination'?

1. What is the definition of average inclination?

Average inclination refers to the average angle at which a celestial body's orbit is tilted in relation to its reference plane. It is typically measured in degrees.

2. How is average inclination calculated?

Average inclination is calculated by taking the sum of all individual inclinations and dividing it by the total number of inclinations. This gives an overall average angle of inclination for the celestial body's orbit.

3. What does a high or low average inclination indicate?

A high average inclination indicates that the celestial body's orbit is highly tilted, while a low average inclination indicates a less tilted orbit. This can have implications for the stability and dynamics of the celestial body's orbit.

4. How is average inclination different from eccentricity?

Average inclination measures the tilt of a celestial body's orbit, while eccentricity measures the shape of the orbit. A high average inclination does not necessarily indicate a high eccentricity, and vice versa.

5. Can average inclination change over time?

Yes, average inclination can change over time due to various factors such as gravitational interactions with other celestial bodies or changes in the celestial body's rotation. However, these changes are typically small and occur over long periods of time.

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