The first version of the qualitative post contained a paragraph at the end in which I did some real math (I have since removed that paragraph, as it appears in a different form in this post). My mother loved the bit about the tennis and thought she had really grasped the general idea; alas, when confronted with a paragraph containing algebraic variables, she felt somewhat bewildered and lost because I hadn't given it enough of an introduction. I was reminded that she hasn't really done any advanced math in several years. Mom, I apologize, and I'm going to take some time now to talk a bit about the philosophy of mathematics in physics because I do plan on using math in this blog whenever it's applicable (which will be, presumably, often, as this is a physics blog).
First, I've titled these two posts as "qualitative" and "quantitative." This is a standard and very useful division in physics. When discussing a subject "qualitatively," one is really trying to grasp the basic idea or gain a sense of intuition. Then, after gaining that first level of understanding, one begins to approach a problem "quantitatively" by actually working the math. I would not be surprised if many of the subjects I broach in this blog will be handled in a similar fashion.
Next, I want to say that one purpose of introducing abstract variables in a mathematical treatment of a problem is to generalize the solution. For example, in the last post, I tried to explain the Doppler effect using a very specific situation. If my mom is hitting tennis balls every two seconds, then decides to run towards the ball machine, she will have to hit tennis balls at a faster rate. We've solved the Doppler effect for that one case. But that situation doesn't apply if we wished to talk about listening to a police siren on a street corner or the rotational speeds of galaxies in an argument about dark matter. That's why the abstraction of math is so useful in physics - we can deal with the problem in such a way that we can use the same language and solution in each instance.
Finally, I'm going to be using variables to represent various parameters in the problem. Specifically, I'm going to use the following:
f0: the frequency at which the balls are emitted by the ball machine
v: the speed or velocity of the tennis balls
vr: the speed or velocity at which my mom runs
f: the frequency at which she hits each tennis ball.
The choice of these variables is completely arbitrary; I could have used anything to represent the quantities of interest. In general, however, we try to use variables that are easily associated with what they represent, like "v" for velocities and "f" for frequencies. The subscripts are then used to delineate different quantities of the same type. f0 is given a "0" because in a sense, it is the original frequency or the initial frequency. My mom's velocity is given the subscript "r" because she is the receiver, in contrast to the source (if the ball machine were moving, I would have described its velocity as "vs").
If there are any questions or comments about this introductory stuff, please do comment below.
Onto the problem. Given the variables defined above, I want to define a couple more terms. If the machine spits out balls at a frequency f0, then the time between each ball, T0, is 1/f0 (in the example, I said that the time between each ball was 2 seconds, so the frequency is then 1/2 per second).
In the time elapsed before the next ball is fired, the previous ball has traveled a distance equal to its speed times the time elapsed, or v*1/f0. This is the distance between each ball.
To determine the time between each ball that my mom observes, we will need the absolute distance between balls (v*1/f0) and the speeds of both my mom and the balls. At the instant when my mom hits a ball, she is v*1/f0 away from the next ball. However, she is still running towards that ball, and the ball is still moving towards her. Therefore, that distance will be covered by the combination of her running towards the ball and the ball moving towards her, i.e. with a velocity equal to the sum of her velocity and the ball's velocity, v+vr. Therefore the time it takes for her to see the next ball is the total distance divided by the total velocity,
T = v * 1/f0 * 1/(v+vr).
To calculate the frequency at which she observes each ball, we invert that time, so
f = (v+vr)/v*f0.
In the example, f0 = 1 per 2 seconds, v = 12 m/s, and vr = 12 m/s, so f = (12+12)/12 * 1/2 = 1 per second, which is exactly what we saw. However, this equation is now general for any similar situation. The equation can be generalized still further to take into account a moving source as well, in which case
f = (v+vr)/(v+vs)*f0.
Now we can talk about listening to sirens on a sidewalk or galactic rotation curves and use the same equation to represent all 3 situations.