On Tuesday, October 18, 2022 at 06:48:56 AM CDT, John Denker via Phys-l <phys-l@mail.phys-l.org> wrote:
In the R&D lab, when encountering a new idea, it is often
fruitful to see how far it can be pushed.
(In the classroom this is not always a good idea, but sometimes.)
In a calculus-based physics course, we can make the problem
of tossing a ball over a building into an amusing optimization
problem. The answer is nontrivial and nonobvious.
You are standing some distance X₀ in front of a building.
The plan is to throw a ball over the building and have it
caught by a second person.
Find the optimal distance between you and the building, so as
to minimize the energy you must put into the toss. Obviously
standing too close or too far away is not good.
The building is 20 meters wide its walls are 25 meters high.
The second person is 40 meters beyond the far side of the building.
Report the optimal distance X₀.
Also report the initial velocity and angle of the toss.
Details:
The ball has a diameter of 7.6 cm.
For simplicity, assume level ground.
For simplicity, assume the profile of the building is rectangular.
In other words, although the roof has "some" pitch, assume it
is not enough to affect the throw.
For simplicity, neglect air resistance and other fluid dynamics.
As always, document any assumptions you make.
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Additional thoughts:
It might make sense to assign the non-optimal version first,
with a fixed specified X₀, asking only for the |V| and angle.
Then the next week, re-assign it asking for the optimal X₀.
Gold star for anybody who accounts for the ball being thrown and
caught at shoulder height (not ground level). That is, encourage
people to think about the real-world situation. Discourage the
notion that anything not mentioned in the statement of the problem
is negligible.
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