Explorations In Engineering: The Daredevil Design Project

This page, while of technical interest, is not required for the execution of the Daredevil Design Project.

There have been several assumptions made with regards to the format of the 'best-fit' curve, and some other mathematical properties of the lesson, which you are here to have explained.

To begin, there is the question of why was .00128*x^2+6.6, a quadratic equation, chosen to fill-in the 'best-fit' curve in DataFlyer? Well, it's not just because it happens to be right for the test data. The initial equations for the marble's run are as follows:

PE_a + KE_a + KE_ra = κ(PE_b)

Where κ is some fraction of the initial potential energy. All energies on the left are subscripted 'after', including the kinetic energy subbed 'rotational'. The potential energy on the right is 'before'. Converting the equations into its longer form:

mgh_l + ½mv² + KE_r = κmgh_i

Now, factor m & g:

h_l + (½v²)/g + KE_r/gm = κh_i

Now, solve for h_i:

h_i = 1/κ(h_l + 1/2g(v_l²) + KE_r/mg)

= 1/2κg(V_l²) + C

With constant C taking over for KE_r/mg, since the rotational energy lost is more or less constant across multiple runs. The rotational loss is really a part of ΔE, the change/loss in energy in the system. Some of that loss is also in the form of sound and heat, but that has been ignored for our purposes. We're not really worrying about the loss due to rotation either, since it is also more or less constant through multiple runs.

Now, you should be able to see where the x² term for the quadratic comes in: from the V_l² term. And, the initial fraction .00128 is the decimal result of 1/2κmg. Knowing that gravity, the marble's mass, and 2 do not change, we could solve for κ. The large constant C turns out to be 6.6 in our test runs.

Now that you have the background for the chosen 'best-fit' terms, you can go back to the web lesson.