My daughter and I were going over science homework this evening. A ball falls for 10 seconds. What is its final velocity?
JC: So how fast is the ball going when it hits the ground?
RC: Zero. It stops before it bounces back up.
JC: Well, how fast is it going just before it hits the ground?
RC: They didn’t ask the almost final velocity. They asked for the final velocity.
13 thoughts on “Final velocity”
Love it — she’s a smart one…
And thus do budding scientists and engineers turn into lawyers.
Very nice! Your mistake – as formulated, the question had nothing to do with hitting the ground (perhaps the ball was dropped 10 feet out of an airplane).
I had this same problem when I was involved in a minor traffic accident. The CHP officer asked me how fast was I going when I hit.
Being a bit dazed, I was genuinely puzzled at the question since I had obviously stopped moving forward. Then I realized the correct answer was how fast was I going just before impact.
Science and mathematics are routinely confused, which does a disservice to our kids’ educations. The “correct” answer is of course the “almost final velocity” since that’s the one based on the simple mathematical model. Your daughter’s answer was the more physically meaningful one. But in a “science” class, that’s incorrect!
At a molecular (and atomic?) level, I’m sure collisions are quite complex and interesting, involving electrostatic forces, properties of the material, vibrations, elasticity, etc. Of course that would be hard to get into in a grade school setting, but isn’t it a shame that it’s ignored in science class?
Nick: Your comment reminds me of a physics lecture explaining why Newton hadn’t discovered conservation of momentum. The instructor threw some putty against the wall and asked us “What happened to all that momentum?” The answer of course is that it’s still conserved in the molecules of the putty, but that is far from obvious and understandably beyond Newton. There’s no conservation of macroscopic momentum, only total momentum.
She is soooo right.
Even a ball that bounces must eventually come to rest. And then its final velocity will be zero.
Keith Devlin wrote a short, insightful article in May last year on how we try to put pure math problems in real world settings in unhelpful ways. He mentions some of the consequences for education, noting that such problems “work provided everyone knows the code and is prepared to play the game,” which younger students are not.
Clarity of thought like RC’s can be a serious liability. My son took at beating in grade 12 calculus. The teacher loved to ask trick questions in the math part of the course, but the applied problems were full of ambiguous questions (unless you DON’T think). One test question I remember is similar to yours: An airplane is going down the runway at 300kph and then rotates to 18 degress, what are the components of motion? I am a physicist, as are 4 coworkers I showed this too. We all said “trick question” with the answer being (300,0), since the plane doesn’t instantly change momentum and there is not enough information for any other physical solution. Apparently this is so wrong that the zero mark could not be appealed. “Correct” answer has the plane rising at 18deg with conserved speed.
Normand: I completely agree. Basic physics is hard if you think about it deeply. If you’re perceptive, you notice more of the implicit approximations and simplifications. You’re not experienced enough yet to understand why they’re justified (or at least pedagogically expedient) but too observant to let them go unnoticed.
Here are a couple other posts I’ve written along these lines, things I wish someone had explained when I was a freshman taking physics classes:
Coming full circle
Infinite is easier than big
Forgive me kicking a dead thread, but the putty momentum example is wrong. Macroscopic kinetic energy is not conserved—the putty heated up a tiny bit from the collision—but macroscopic momentum is conserved.
The putty’s momentum first went into the wall, which was very briefly accelerated and deformed (perhaps only microscopically) by the impact (before bouncing back). The momentum ultimately propogated through the whole earth (everywhere propogating at the local speed of sound). The resulting net acceleration of the earth was undetectable because of the earth’s great mass. This acceleration of the earth is almost exactly the vector opposite of the acceleration of the earth earlier caused by the instructor throwing the putty. The putty merely borrowed some momentum from the earth.
The page with Keith Devlin’s article in Scott’s comment (http://www.maa.org/devlin/devlin_05_10.html) is not working… Anyone?