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!