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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.

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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.

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  • $\begingroup$ 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. $\endgroup$ Commented Jan 24, 2022 at 11:37
  • $\begingroup$ @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) $\endgroup$ Commented Jan 24, 2022 at 12:05
  • $\begingroup$ 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? $\endgroup$ Commented Jan 24, 2022 at 12:39
  • $\begingroup$ Complete transfer of energy on impact yes $\endgroup$ Commented Jan 24, 2022 at 13:05
  • $\begingroup$ 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 $l = B \sin(\pi/4) = B / \sqrt{2}$ $\endgroup$
    – SrJaimito
    Commented Jan 24, 2022 at 16:00

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