If a person can jump a horizontal distance of 3.0 m
on Earth, how far could the person jump on the
moon, where the free-fall acceleration is g/6 and
g=9.81 m/s^2? How far could the person jump on
Mars, where the acceleration due to gravity is 0.38g?
When I first saw that, I thought it was nice. The physics
of jumping is something the students can relate to. This
does not even require going to the moon, because you can
simulate reduced gravity using a Peter Pan flying-wire rig.
A moment later I thought "Oh no. I betcha the book is expecting
answers based on wildly wrong physics." And sure enough, the
answers in the back of the book are 18 m and 7.9 m. In other
words, the book assumes the length of the jump scales inversely
with the gravitational acceleration. This is quite wrong.
It seems to me that this exercise promotes the worst sort
of equation-seeking and plug-and-chugging. Find the formula
for how far a bullet flies and plug in a non-standard value
of g.
I give the book some small credit for occasionally reminding
students to check their work ... but there aren't enough such
reminders ... and what's worse, they seem to be mostly in the
context of equation-seeking:
-- Check that you plugged the right numbers into the equation,
rather than:
++ Think about the physics, and see if it is consistent with
common sense.
Let's try applying a little common sense to the jumping question.
The book asks us to consider three cases: normal 1-Gee gravity,
1/6th Gee, and approximately 1/3rd Gee. I suggest we should also
consider an _increase_ in gravity. Under 3 Gee conditions, an
ordinary person can't even stand up, let alone walk, much less
jump a distance on the order of 1 meter. So ... the formula the
book wants us to use cannot possibly be valid for variable-g
jumping.
I don't expect high-school students to figure out on their own
exactly /why/ the obvious approach is wrong, or how to fix it ...
but they should be able to notice that it is clearly, seriously
wrong.
We're not students, so let's discuss amongst ourselves. Actually,
it's not very hard to figure out what's going on. The book is
evidently assuming the energy of the jump is independent of the
local gravity. This cannot possibly be true. Leg muscles produce
a limited amount of force. If this force is not enough to overcome
body weight, you cannot jump at all. So the simplest model that
makes any sense at all would involve something like
E(jump) = constant - m g(local) s [1]
where s represents the "stroke" i.e. the distance you extend your
leg during the jump, something on the order of 1 meter.
It's easy to see why the "m g s" term is missing from the usual
"ballistic range" equation:
++ The height to which you can shoot a bullet is enormous
compared to the size of the gun.
++ The height to which you can throw a baseball is reasonably
large compared to the length of your arm.
-- In contrast: The height to which you can jump is *not* large
compared to the length of your legs. (The world-record high-jump
is only on the order of 2 meters, and the exercise in question
involves a jump even weaker than that.)
So, for bullets and baseballs, the "m g s" term in equation [1] can
be neglected to a decent approximation, whereas for jumping, that
term is a huge correction. It is on the order of 100% of E(jump).
You could build an even fancier model than equation [1], perhaps by
including the force/velocity tradeoff that real muscles exhibit ...
but that's not the point. We don't need a quantitative model to
know that the constant-energy model is wrong. It's qualitatively
wrong, wildly wrong.
Here's the real point: IMHO one of the most fundamental goals of
the whole education system is to teach kids how to *think* clearly.
In computer class, one of the rules for debugging a program is to
check a few typical cases, then check the corner cases, and so on.
It seems to me that science class ought to be a least as scientific
as computer class! Before using an equation, students should be
expected to *debug* the equation, by testing it against a few cases
where the answer is known.
Note the contrast:
-- If I find a typo in the text, it doesn't bother me. I hardly even
notice.
-- If I find wrong physics in the text, that's annoying, but not worth
making a fuss about. Mistakes happen. I've made plenty of mistakes.
++ When I find wrong physics *that should have been easily caught* it
bothers me, insofar as it indicates a lack of thinking ... or more
specifically, a lack of the /right kind/ of thinking.
Equation-seeking is not science. I don't know what it is, but it's not
science. Teaching students to think about the real physics is important,
but it's hard ... especially when the textbook is more-or-less explicitly
rewarding them for mindless equation-seeking.
Some of the equations of physics work amazingly well: For example,
E = G M m / r works on laboratory length-scales and cosmic length-scales
and everything in between. In contrast, E = m g h works to first order
on laboratory length-scales and does /not/ generalize at all well. So
it pays to check each equation before using it.
The funny thing is, qualitatively checking the physics is often /easier/
and more fun than doing the algebra. It involves more imagination and
creativity. In the case of the ballistic range equation, almost any student
can appreciate the idea of simulating "extra" gravity by putting somebody
in a harness and using bungee cords to apply several hundred pounds of
downward force ... i.e. a reverse Peter-Pan rig ... whereupon they're not
going to be doing much long-jumping.
Of course, an ordinary, upward, gravity-reducing Peter Pan rig is more
interesting.
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Possibly constructive pedagogical suggestion: Set up a program to reward
students for finding things in the textbook that cannot possibly be correct.
They should be encouraged to look for genuinely wrong physics (as opposed
to mere nit-picking).