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I'm looking for a good example for a freshman mechanics class to demonstrate how one can integrate the equations of motion numerically when there is no closed-form solution. The problem below is the best I've been able to come up with yet, but I'm not totally happy with it, and I'm wondering if anyone can suggest a better one. Here are my criteria:
(1) It should be of the form where essentially the acceleration (or force) is given as a function of time from t1 to t2, and we want to integrate to find either v or x. It's also OK if the force is velocity-dependent.
(2) It should be physically simple, and it should not depend on any knowledge of vectors, two-dimensional motion, energy, or Newton's law of gravity. Preferably it would not even assume any previous knowledge of topics like static and kinetic friction.
(3) There should be no closed-form solution in terms of elementary functions like exponentials, trig functions, and logs.
(4) It should be a problem that is physically interesting, natural, and well-motivated -- not just something where I write down some expression that happens to be impossible to integrate in closed form.
(5) I would also like to be able to come up with a second variation on the same problem where we want to find the time at which x or v reaches some value of interest, and I would like this variation to be physically interesting and well motivated.
My Antarctic rescue example below fails criterion 2, mainly because it involves two-dimensional motion.
I'd be grateful for any suggestions!
BTW, if you want to figure out whether a function is not integrable in closed form, an easy way to do it is to go to integrals.com and type it in.
A really natural example is the motion of a falling object subject to a force from air friction that is proportional to v^2. However, this example fails criterion #3, since there is a closed-form solution (although it is not easy to find).
Thanks in advance!
-Ben
==============
The pilot of a small airplane crashes in Antarctica in a storm and activates her personal locator beacon.
A rescue helicopter is unable to land due to whiteout conditions near the ground, but will drop
a supply package to her known location from the minimum height yo=30 m to which it can safely descend.
The package is blown sideways by the wind, so the helicopter, hovering at height yo,
needs to release it from a point displaced by a horizontal distance x.
Based on experiments by Von K\'arm\'an (1881-1963),one typically
expects that wind speed is proportional to [itex]y^{1/7}[/itex], and one also usually finds that air friction
depends on the square of the speed, so that the horizontal force on the package is given
by [itex]ky^{2/7}[/itex], where k is a constant. Let k=10 in SI units, and let m=1 kg. Find x.
The vertical motion is known in closed form,
[tex]
y = y_\zu{o}-\frac{1}{2}gt^2 \qquad .
[/tex]
We then have a horizontal acceleration given by
[tex]
a = \frac{F}{m} = \frac{k}{m}\left(y_\zu{o}-\frac{1}{2}gt^2\right)^{2/7} \qquad .
[/tex]
This acceleration will be applied from t=0 to [itex]t=\sqrt{2y_\zu{o}/g}[/itex].
(1) It should be of the form where essentially the acceleration (or force) is given as a function of time from t1 to t2, and we want to integrate to find either v or x. It's also OK if the force is velocity-dependent.
(2) It should be physically simple, and it should not depend on any knowledge of vectors, two-dimensional motion, energy, or Newton's law of gravity. Preferably it would not even assume any previous knowledge of topics like static and kinetic friction.
(3) There should be no closed-form solution in terms of elementary functions like exponentials, trig functions, and logs.
(4) It should be a problem that is physically interesting, natural, and well-motivated -- not just something where I write down some expression that happens to be impossible to integrate in closed form.
(5) I would also like to be able to come up with a second variation on the same problem where we want to find the time at which x or v reaches some value of interest, and I would like this variation to be physically interesting and well motivated.
My Antarctic rescue example below fails criterion 2, mainly because it involves two-dimensional motion.
I'd be grateful for any suggestions!
BTW, if you want to figure out whether a function is not integrable in closed form, an easy way to do it is to go to integrals.com and type it in.
A really natural example is the motion of a falling object subject to a force from air friction that is proportional to v^2. However, this example fails criterion #3, since there is a closed-form solution (although it is not easy to find).
Thanks in advance!
-Ben
==============
The pilot of a small airplane crashes in Antarctica in a storm and activates her personal locator beacon.
A rescue helicopter is unable to land due to whiteout conditions near the ground, but will drop
a supply package to her known location from the minimum height yo=30 m to which it can safely descend.
The package is blown sideways by the wind, so the helicopter, hovering at height yo,
needs to release it from a point displaced by a horizontal distance x.
Based on experiments by Von K\'arm\'an (1881-1963),one typically
expects that wind speed is proportional to [itex]y^{1/7}[/itex], and one also usually finds that air friction
depends on the square of the speed, so that the horizontal force on the package is given
by [itex]ky^{2/7}[/itex], where k is a constant. Let k=10 in SI units, and let m=1 kg. Find x.
The vertical motion is known in closed form,
[tex]
y = y_\zu{o}-\frac{1}{2}gt^2 \qquad .
[/tex]
We then have a horizontal acceleration given by
[tex]
a = \frac{F}{m} = \frac{k}{m}\left(y_\zu{o}-\frac{1}{2}gt^2\right)^{2/7} \qquad .
[/tex]
This acceleration will be applied from t=0 to [itex]t=\sqrt{2y_\zu{o}/g}[/itex].