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

[newjerseyrunner]

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.


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.

 

[mfb]

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.