The new academic year is starting, and if there's one thing students love it's a good word problem. *If Sue is four times as old as John will be when Sue is one year than John...* So in that spirit I was amused to find basically this kind of problem in a college physics textbook I was perusing for post ideas as I get back into the swing of blogging. It runs thusly:

A father racing his son has half the kinetic energy of the son, who has half the mass of the father. The father speeds up by 1.0 m/s and then has the same kinetic energy as the son. What are the original speeds of the father and the son?

You don't really need to know any physics to do this, other than just the equation for kinetic energy. The kinetic energy of an object of mass m and velocity v is:

All right, give it a shot. Click below for the solution! (Seriously, if you have math or intro physics classes this semester it's a good warm-up)

My guess is that the reason people hate these so much is that if you either do the guess/check method or just think really hard you can often get the solution. But really you don't have to do either, the point is to write the problem algebraically and then just turn the crank. That first step is what people have trouble with sometimes.

So initially the father has half the kinetic energy of the son. Using notation that I trust is self-explanatory, this is just:

And the son has half the mass of the father:

So we can write down the kinetic energy equations explicitlyl:

Where on the right side the son has half the mass of the father (one of the 1/2s) and the father has one half the kinetic energy of the son (the other 1/2). Let this rat's nest of parentheses cancel and take the square root:

So initially the son is running at twice the speed of the father. That's a good thing to know, but it doesn't quite answer the problem yet. Now we also know that when the father speeds up by 1 m/s, the kinetic energies are equal:

Plugging in the stuff we already figured out:

Solve that with the quadratic formula, you get that the initial velocity for the father is 2.4 m/s. The son's velocity was (as we saw) twice that, so he was going 4.8 m/s. Yay for math!

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Yah for math, but boo to the physics behind this particular problem. It's difficult to see how such a contrived calculation could ever be useful in a real-life situation involving measurable quantities. Conservation of energy is a very useful principle, but it really doesn't make any sense to equate kinetic energies of the two runners (how would you know that, anyway, without measuring their speeds, or slamming them into a calorimeter). Introductory physics textbooks should involve problems where all the inputs are actually measurable, and the outputs are actually interesting to calculate. Some nice examples are in Riley "Problems for Physics Students", which you can find parts of on Google books.

I agree entirely. It is kind of entertaining to work through, even in it doesn't make a good physics problem. Still, it's an even-numbered problem from a very mainstream text (Halliday & Resnick) so some students are going to be forced to work through it.

You know how football players do press-ups in their training? Have you ever seen, in any football game, any of the players lay down mid-game and do press-ups? Then why do they bother to do it in training? It's to strengthen their muscles, which they use in other ways. Same here; you'll never be able to directly measure the kinetic energy of a runner, as you say, but the algebra and thinking processes behind the reasoning are used in other ways. This is just a contrived, yet simplified, way of strengthening your skills. That's what I tell my students, anyway, when they ask, "Why are we doing this? I'm never going to be a scientist". I hope it helps. :-)

At the risk of being overly pedantic, there are actually two solutions here -- v_f could either be 2.414 m/s or -0.414 m/s, either of which could be physically meaningful. Chasing a 3 year old around is a lot easier at 0.4 m/s.

Agree completely with #1. A nice conservation of energy and momentum problem would be quite interesting if it's intuitive.

I hated wordy problems when I was at school, took me ages to get my brain around them even when they were easy. Wordy geometry problems were the biggest killer.

I like your "bench press" analogy. A much easier solution is that the father's K doubles w/ a increase in velocity of 1. So for the father, .5m(v+1)^2 = 2(.5mv^2). No need to involve the son's mass or velocity. I'm also interested if mrgeek chases 3 year olds walking backwards?

Longhorns Rule!

mrgeek, that may work out in pure math, but the problem said that the father was racing the son. It doesn't make much sense that the father would start out running in the opposite direction. "Speeds up by 1.0 m/s" implies that the acceleration is in the same direction as the initial velocity, too.

Whoops, I forgot that the son's velocity might be negative too. In that case the father starts out racing him and ends up running in the opposite direction, which isn't ruled out by the text. My second objection stands regardless, though.

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