# Maximum theoretical displacement from a ruler catapult

I am considering giving my students the following problem:

Assume we have a ball placed upon a ruler, residing on a table. Now we drop another ball from an height $$h$$ onto the edge of the ruler. If $$L$$ is the length of the ruler and $$\ell$$ is the length of the ruler outside of the book then $$x = \ell /L$$.

Assuming no drag, what is the optimal value for $$x$$ to maximize the displacement $$d$$?

Second assumption assume that $$x$$ is chosen such that the catapult does not hit the book edge upon rotation / that this hit, does not inhibit the launch.

I want my students to do the experiment physically, take measurements, make sure to keep the height $$h$$ consistent etc. However, I sadly lack the formal background to compute the theoretical ideal values. I can see that the angle depends on the inertia, and if we knew the exit angle, we could compute the ideal value for $$x$$ from the projectile equations, but this is how far I've gotten

\begin{align*} x &= v_0 t \cos \alpha \\ y &= v_0 t \sin \alpha - \frac{1}{2}gt^2 + B \end{align*}

With $$v_0 = \sqrt{2gh}$$.

Any help is appreciated. Hopefully this problem has a non trivial solution, which would make it interesting for the students to work out by experimenting.

EDIT: I am sort of not interested in the exact solution here but more the underlying principles. Or rather, I am interested if this problem has a "trivial" solution experimental wise, or if it will lead the students to an interesting optimization problem.

• Please give an idea what is $x = \frac{l}{L}$ so that I can get your question clearly and help you to get out of this problem. Commented Jan 24, 2022 at 11:37
• @TejasDahake $x = \ell / L$ is the ratio of the ruler outside of the table. So $0 < x < 1$, or am I missing something? I would guess there are real world restrictions imposed so that maybe $1/4 < x < 3/4$, but does such restrictions has to be taken into consideration when developing our theoretical model? (Sorry for not being a physisist) Commented Jan 24, 2022 at 12:05
• Are you assuming that the velocity of ball which is under the free fall will loose its complete energy before touching the surface of the table? Commented Jan 24, 2022 at 12:39
• Complete transfer of energy on impact yes Commented Jan 24, 2022 at 13:05
• Probably I didn't understand the proposed system very well. However, if you keep $h$ constant through experiments, the modulus of the velocity of the ball leaving the ruler should be almost the same, and the factor that becomes relevant for the distance $d$ is the angle at which the ball leaves the rule. Maximum distance for a parabolic shot is achieved for 45 degrees, and you get this when $l = B \sin(\pi/4) = B / \sqrt{2}$ Commented Jan 24, 2022 at 16:00