I noticed a very important lesson in Rhett’s post July 4th and an example of work-energy: checking the answer.

Let me make some checks here. Will the acceleration be positive? Yes. The first term will always be positive and greater than g because (d+h)/d is greater than 1. What if a jumper jumps from a higher height (h). This would make the acceleration greater. What if the jumper stopped in less water, this would also make the acceleration greater. Finally, does this have the correct units? Yes.

Notice how Rhett isn’t checking the numerical answer — this is a check of the equation that leads to the numerical answer, to see if it’s reasonable. There’s a lot of power in doing this that one loses when the numbers are substituted too early in the process. While you can do the first and last checks — direction and units — the trends of what happens if a variable changes is removed. And checking the limiting behavior of an equation is a tremendously important tool as the questions, and resulting conclusions, get more complex.

Now, once you get the answer, you can check that for reasonableness, too. As I mentioned some time back, when I taught we stressed getting answers that made physical sense, else your math mistake be tagged as a conceptual error. You should not be deducing that a frog has a mass of 10^24 kg; one can check this by relating the mass to known objects (in this case, being a measurable fraction of the mass of the earth) which requires having some awareness of masses (or forces, energy, etc.) on different scales. Or can apply the long-lost art of estimating the answer from the number you put in. All numbers become 1,2 or 5 and you round aggressively — but the rounding often cancels, and you can get pretty close. At least close enough to be within a factor of 10 or less of the right answer.

And some of this is shown in sciencegeekgirl’s Teaching the gentle art of estimations which includes a simple estimation problem which was a complete disaster when asked of some teenage students.

The conclusion I draw from this? We’re doomed.

Interestingly (ironically?) the very last example has a number I question. The force of impact of an object falling under the influence of gravity is much larger than mg.

Acceleration goes as the velocity of impact divided by the time of contact. What is the time of contact? The bottom of the ball hits the ground, but the top keeps going until it gets the signal that the bottom has hit, that there’s no more room to move down, and it’s time to start moving up. That happens at the speed of sound.

And from that, a time of 10 microseconds is concluded, giving a force of 10,000 mg (i.e. an acceleration of 10,000 g’s. Wow!)

I balked at that (and commented in the post). The object doesn’t recoil that fast — that’s the limiting case for the top to know that there has been an impact. One needs to look at the spring constant of the material to know what’s going on. 10 microseconds is too short — the contact time is almost certainly much longer. How much longer, I wonder? A convenient scale with which I am becoming familiar is shutter speed. 10 microseconds wouldn’t be discernible on a high-speed camera, if I had access to one. Which I do.

This is at 420 frames/second, and since I have the advantage of being able to easily click through frame-by-frame on the original, I’ll tell you the answer: the ball is in contact for ~4 frames, or just under 10 milliseconds. IOW, almost three orders of magnitude longer than the speed-of-sound estimation.