Creating a function with specific shape, intercepts, integral....

In summary, the conversation discusses the process of calculating the peak draw weight of a bow based on the draw length, velocity of the arrow, and the known shape of a curve. The function must intersect the x-axis at the draw length and the y-axis at a known scalar for the peak let-off. The integral of the function from 0 to the draw length should accelerate the arrow to a known velocity. However, there is a circular dependence as the function is unknown but is needed for the integral calculation. The suggested solution is to write down the function with the peak weight as an unknown parameter and calculate the integral, setting it equal to the known arrow energy. Additional assumptions may be needed to reduce the problem to one unknown.
  • #1
newjerseyrunner
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I'm trying to see if I can calculate the peak draw weight of my bow based on the draw length and the velocity of the arrow and a known shape of a curve, but I'm not quite sure how to make such a function, if there even is such a way.

compound-draw-force-curve.jpg
This is the shape of the draw weight plotted against distance, so the the force applied to the arrow is this same shape, reflected over the y-axis and translated by the arbitrary but known draw length L.

The function must have intersect the x-axis at L because that is when the arrow releases and no further force is applied to it.

The function also needs to intercept the y-axis at some arbitrary but known scalar to the peak Letoff. My particular bow has a 70% setoff, so the x value of the y intercept should be 30% of whatever the peak is.

The integral of the function from 0 to L should accelerate the arrow of an arbitrary but known weight W to an arbitrary but known flight velocity V. Keeping in mind that L is a unit of draw length, not time, which makes this even more puzzling for me. If I know the final velocity of the arrow and the total distance it accelerated for, I feel like I should be able to figure out a time scalar for L.

But I also feel like I got stuck with some circular dependence: I seem to need to be able to do an integral on a function that I don't know yet.

Am I hopelessly stuck, or might someone point me in the right direction? I feel like knowing the shape of the plot, it's intercepts, known flight speed, and distance of acceleration should allow me to calculate backwards and get the peak draw weight, but everything is so interconnected and I come up with a curve that has the intercepts correct for my bow, but it's not general because changing variables changes the intercepts.Or I could buy a gauge and measure it, but I feel like this should be doable.
 
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  • #2
In practice some energy will go into the bow, but it might be a good first approximation that all energy goes into the arrow. The total energy is simply the integral of the curve over the full draw profile. You know the energy of the arrow.

Write down the function with the peak weight as unknown parameter, calculate the integral (it will depend on the peak weight), set it equal to the known arrow energy.
If you don't know the shape of the curve well enough you'll need to make some additional assumptions about it to reduce the problem to one unknown.
 

Related to Creating a function with specific shape, intercepts, integral....

1. How do I create a function with a specific shape?

To create a function with a specific shape, you will need to determine the equation that represents that shape. This can be done by analyzing the characteristics of the shape, such as its intercepts, slope, and curvature. Once you have determined the equation, you can use it to create your function.

2. How do I find the intercepts of a function?

The intercepts of a function are the points where the function crosses the x-axis or the y-axis. To find the x-intercepts, set the y-value of the function to 0 and solve for x. To find the y-intercepts, set the x-value of the function to 0 and solve for y. These points will be the intercepts of the function.

3. How can I ensure that my function has a specific integral?

The integral of a function represents the area under the curve of the function. To ensure that your function has a specific integral, you will need to determine the limits of integration and the value of the integral. You can then use this information to adjust the equation of your function to achieve the desired integral.

4. Can I create a function with multiple shapes and intercepts?

Yes, it is possible to create a function with multiple shapes and intercepts. This can be done by using different equations for different parts of the function, or by using a piecewise function. By combining multiple shapes and intercepts, you can create a more complex and versatile function.

5. What are some common techniques for creating a function with specific characteristics?

Some common techniques for creating a function with specific characteristics include using transformations, such as shifting, stretching, or reflecting the graph of a basic function. Another technique is to use the properties of functions, such as even or odd functions, to create the desired shape. Additionally, using mathematical operations, such as addition, subtraction, multiplication, and division, can help achieve the desired characteristics of a function.

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