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Re: [Phys-L] Freshman Physics / toss ball over building



On 10/14/22 3:32 AM, Peter Schoch wrote:

1. You are standing 20 meters from a building. The building is 10 meters
wide and 35 meters high. The second person is 50 meters from the other
side of the building. With what initial speed and launch angle should a
ball be thrown to get to the second person.

HINT: The “peak” of the building (35 m) is located directly at the
midpoint of the building. To ensure that the ball clears the top of the
building, its height above the ground should be no less than 35.1m at this
point.


0) As Anthony Lapinski pointed out, there are numerous deeper
problems we ought to discuss. However, I haven't got time for
that right now, so let's focus on the narrow question above.

1) IMHO this is a hard question by HS standards, but not
insanely so.

Specifically, it should be OK to assign such a problem ...
/provided/ the proper foundation has been laid. So IMHO the
issue reduces to /how/ to lay the proper foundation.

a) Nowadays students are fixated on standardized tests that
have harsh time pressure and equal points for each question.
As a consequence, they have been trained that any question
that cannot be solved in a single step is not worth solving.

So by way of preparation, it should be explained that this
is not an SAT question. The solution requires multiple steps.
None of the steps is hard, but you do have to proceed one
step at a time.

b) Draw the diagram. I can't tell you how many times I've seen
beginners try to solve a problem in their head in situations
where world-class experts would draw the diagram.

https://av8n.com/physics/img48/toss-over-building.png

c) Suggested refinement: I would drop the word "HINT". To my
ears, a hint is optional information, and it should be possible
to solve the problem without any hints. In this case the second
paragraph contains indispensable additional information.

-- The ".1" in the critical height "35.1" comes out of nowhere.
Working backwards one can infer that the ball has a radius of
.1 m, but only after seeing the hint.

Note that a baseball has a radius of about 0.038 m. A ball
3× bigger than that might be awkward to throw, and/or might
have an unduly low density.

d) Similarly, the stipulation that the critical height occurs
at the midpoint comes out of nowhere. Working backwards (after
seeing the hint) we can infer that the building must have a
steeply pitched roof, on the order of 45° or more, with no eaves,
as you can see in my diagram. Otherwise (with a flatter roof
and/or eaves), a steeper throw and a considerably greater |V|
would be needed to clear the near-side edge of the building (as
suggested by the dotted line in the diagram).

A 45° pitched roof might make sense for shedding snow in Buffalo
NY, but not in Tucson AZ.

I don't immediately see any authentic way to incorporate the idea
that the peak of the roof is the only thing that matters; not
without adding to the complexity. Therefore I would be tempted to
either:
-- replace the "building" with a /wall/ and tell people to ignore
the thickness of the wall, or
-- state that the roof has a very *small* pitch and no overhanging
eaves. This changes the physics of the situation. The midpoint
is no longer critical. It seems to me that the students would
learn just as much from this modified version.

e) Suggested refinement: Since the questions specifies "thrown", I
would decrease the heights and distances to bring the required |V|
more into line with what a HS athlete could achieve.

IMHO it is important when constructing rich-context problems to
make sure that the details check out, so that students are not
penalized for bringing real-world facts to bear.

Also it might be good to eliminate the word "launched" in favor of
/thrown/, for consistency.

And dial down the radius, as mentioned above.

And specify that air resistance and other fluid-dynamic effects
are to be neglected. Otherwise it is waaaay too hard.

f) Suggested refinement: I suspect the question intended to ask for
the minimum-|V| or minimum-energy solution. If so, this should be
stated.

Otherwise, the problem is ill-posed ... which might be a good thing.
Specifically, any overestimate of the angle is a valid answer to
the question. For example, a qualitative sketch tells me that 80°
is an overestimate, whereupon the exact height of the building no
longer enters the calculation.

Remember: As I have said again and again: Whenever you encounter a
new problem, start by asking how badly ill-posed it is.

This illustrates what I mean by laying the foundation: If students
have been taught to watch out for ill-posed problems, they should
be able to notice the non-minimal solutions.

This is one of eleventeen reasons why you should draw the diagram
already. See item (b) above.

g) I would be sorely tempted to specify a building that is just
barely too short, so the minimum-energy solution clears the
building with some small amount of room to spare, and see if
anybody notices.

h) As another example of ill-posedness, the height of the thrower
and catcher are not specified. It seems rational to surmise that
the ball is thrown and caught at shoulder level, not ground level.
In this case, a good student might just pick a reasonable shoulder
height (e.g. 1.5 m), document the assumption, and proceed from there.
Also assume level ground and /document the assumption/.