Every section of Physics 218 I’ve taught this semester has asked me about this question. Really it’s less of a physics question than it is a math question, but either way it gives people fits. It’s not all that surprising. While it seems like it should be simple, to most beginning students it’s not at all intuitive how all the given information gits together. Without further ado, the problem:
Earthquakes produce several types of shock waves. The most well-known are the P-waves and the S-waves. In the earth’s crust, the P-waves travel around 8.9 km/s while the S-waves move at about 2.7 km/s. The time delay between the arrival of these two waves at a seismic recording station tells geologists how far away the earthquake occurred. If the time delay is 14 s, how far away from the seismic station did the earthquake occur?
Numbers are for suckers. Call the faster speed v1 and the slower one v2. Call the time delay T. Call the unknown distance d. Call the time between the earthquake and the first wave t1, and the time between the earthquake and the second wave t2.
We know v1, v2, and T. We don’t know d, t1, or t2.
Why did we do all that defining? If you don’t know where to start, the first thing to do is to think systematically about what you know and what you don’t. That might give you an idea as to how you might proceed. Distance is velocity times time, so this is true:
Which is nice, but not enough. There are three unknown quantities and two equations.
As a rule, if you have a certain number of unknown quantities you need that same number of equations to uniquely specify all of them – or to guarantee that there is no consistent answer. To be fair the mathematicians can tell you a whole stack of caveats to that statement, but they essentially boil down to requirements that your equations also not be mere restatements of each other.
So we need a third equation. We happen to have one, hidden in the fact that we know the time delay:
It doesn’t matter which order we solve these three equations in, so the following is one of several paths leading to the answer. Take this T equation, solve it for t2, and plug it into the second of our d equations. You’ll get:
Take that, solve it for t1, and plug it into our first d equation. You’ll get:
We’re almost done because that equation only involves d and the things we’re given. So solve it for d.
Now we’re done. The longer the time delay, the farther the quake is. For very large v1, the distance is very close to v2 t2 as we expect. Nice to see that the limiting cases check out.
Is it very difficult, once you know how to do it? Of course not. But it is great practice in recognizing equations you may not immediately know you have.