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# Math Question: What is the max distance of x?

Posted on 2016-11-08
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What is the maximum distance of x in order to limit the slope change to less than .15 degrees?
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Question by:Precision

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Expert Comment

ID: 41879340
It depends.
What is the curve?  A circle or a parabola or something else?
Are you launching a rocket or a car?
Are you shooting a bullet or a canon ball?  How fast is it going at the start?
And did you really mean 0.15 degrees?  That is such a small angle that it is difficult to measure it by hand.

The slope is slightly positive to the left at -1.  It is slightly negative to the right at +1.
A change of 0.15 degrees could mean that it starts at +0.075 degrees and changes to -0.075 degrees.  Is this what you want?
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Assisted Solution

John Hurst earned 83 total points
ID: 41879379
Is this the railroad track question ?

You can readily approximate by assuming you have two back to back triangles and work it out that way.

For a mile long railroad track with 1 foot inserted, x is about 44 feet
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Assisted Solution

Jeff Darling earned 83 total points
ID: 41879397
.0026
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Accepted Solution

d-glitch earned 251 total points
ID: 41879409
If the curve is a circle, and if the angles are x = ±0.075 deg,
then the radius of the circle is  R = 1/sin(x) = 763.9439
and the height at the center is  R(1 - cos(x)) = 0.0006545
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BigRat earned 83 total points
ID: 41880361
0.15 degrees is pi/1200 radians. Since the slope of a tangent to any curve is dy/dx = tan(theta) then theta must be less than pi/1200. The value of x needed to be found is obtained by setting y to zero. Therefore we need two conditions y=0 and dy/dx to be less than tan(pi/1200) - at least in the range [-1,+1]. So what we now need to know is what y is as a function of the abscissa.

Given the curve is a circle. First the circle must cut the abscissa at -1 and +1 which means it is symmetric in x so that the equation is x² + (y+a)² = r² where r is the radius and a the y offset to the center. Since the center is below the x axis a must be negative. We immediately obtain the result (y+a)²=r²-1. Similarly we obtain dy/dx = -x/sqrt(r²-1) (by substituting (y+a) after differentiation. This must still be less than tan(pi/1200) but we still have TWO unknowns r and a.
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