I am a formula snob.

We all know about grammar snobs: the ones who complain bitterly about people using *who* instead of *whom*. Many people know how to use *whom* correctly; only grammar snobs *care* about it. I gave up the *whom* fight long ago (let’s just let *whom* die) but I am a grammar snob when it comes to certain words. For example, ‘til is *not* a word, as I have discussed before.

However, I am almost always a formula snob.

Consider this formula from the text I’m currently using in freshman physics:

x = v_{0} t + ½ a t^{2}.

To me, looking at this equation is like watching Miley Cyrus twerk with Beetlejuice. I would much, much rather the equation looked like this:

Δx = v_{0} Δt + ½ a Δt^{2}.

The difference between these two formulas is profound. To understand the difference, we need to talk about positions, clock readings, and intervals.

A position is just a number associated with some “distance” reference point. We use the variable *x* to denote positions. For example, I can place a meter stick in front of me, and an ant crawling in front of the meter stick can be at the position *x*=5 cm, *x*=17 cm, and so on.

A clock reading is just a number associated with some “time” reference point. We use the variable *t* to denote clock readings. For example, I can start my stopwatch, and events can happen at clock readings *t*=0 s, *t*=15 s, and so on.

Here’s the thing: physics doesn’t care about positions and clock readings. Positions and clock readings are, basically, arbitrary. A football run from the 10 yard line to the 15 yard line is a 5 yard run; going from the 25 to the 30 is also a 5 yard run. The physics is the same…I’ve just shifted the coordinate axes. If I watch a movie from 8pm to 10pm (say, a Matt Damon movie) then I’ve used up 2 hours; the same thing goes for a movie from 9:30pm to 11:30pm. Because a position *x* and a clock reading *t* ultimately depend on a choice for coordinate axes, the actual values of *x* and *t* are of little (physical) importance.

Suppose someone asks me how far I can throw a football. My reply is “I threw a football and it landed on the 40 yard line!” That’s obviously not very helpful. A single *x* value is about as useful as Kim Kardashian at a barn raising.

Or suppose someone asks, “How long was that movie?” and my response is “it started at 8pm.” Again, this doesn’t say much. Physics, like honey badger, doesn’t care about clock readings.

Most physical problems require *two* positions, or *two* clock readings, to say anything useful about the world. This is where the concept of *interval* comes in. Let’s suppose we have a variable Ω. This variable can stand for anything: space, time, energy, momentum, or the ratio of the number of bad Keanu Reeves movies to the number of good (in this last case, Ω is precisely 18.) We define an interval this way:

ΔΩ = Ω_{f} – Ω_{i}

So defined, ΔΩ represents the *change* in quantity Ω. It is the *difference* between two numbers. So *Δx = x _{f} – x_{i}* is the

*displacement*(how far an object has traveled) and

*Δt = t*is the

_{f}– t_{i }*duration*(how long something takes to happen).

When evaluating how good a football rush was, you need to know where the player started and where he stopped. You need two positions. You need *Δx*. Similarly, to evaluate how *long* a movie is, you need the starting and the stopping times. You need two clock readings. You need *Δt*.

I’ll say it again: most kinematics problems are concerned with *Δx* and *Δt*, not *x* and *t*. So it’s natural for a physicist to prefer formulas in terms of intervals (Δx = v_{0} Δt + ½ a Δt^{2}) instead of positions/clock readings (x = v_{0} t + ½ a t^{2}).

But, you may ask, is the latter formula *wrong*?

Technically, no. But the author of the textbook has made a choice of coordinate systems without telling the reader. To see this, consider my (preferred) formula again:

Δx = v_{0} Δt + ½ a Δt^{2}.

The formula says, in English, that if you want to calculate how far something travels *Δx*, you need to know the object’s initial speed *v _{0}*, its acceleration

*a*, and the duration of its travel

*Δt*.

From the definition of an interval, this can be rewritten as

x_{f }– x_{i} = v_{0} (t_{f }– t_{i}) + ½ a (t_{f }– t_{i}) ^{2}.

This formula explicitly shows that *two* positions and *two* clock readings are required.

At this point, you can simplify the formula if you make two arbitrary choices: let x* _{i}* = 0, and let

*t*= 0. Then, of course, you get the (horrid) expression

_{i}x = v_{0} t + ½ a t^{2}.

I find this horrid because (1) it hides the fact that a particular choice of coordinate system was made; (2) it over-emphasizes the importance of positions/clock readings and undervalues intervals, and (3) it ignores common sense. Not every run in football starts at the end-zone (i.e. *x *= 0). Not every movie starts at noon (i.e. *t* = 0). The world is messier than that, and we should strive to have formulas that are as general as possible. My formula is *always* true (as long as *a* is constant). The horrid formula is only true *some* of the time. That is enough of a reason, in my mind, to be a formula snob.

**Bonus exercise**: show that the product

Ω_{Keanu Reeves} x Ω_{Matt Damon} ≈ 3.0

has stayed roughly constant for the past 15 years.

on September 17, 2013 at 2:07 PM |AnonymousHoney Badger definitely does not care!

on October 8, 2013 at 2:57 PM |AnonymousI guess Omega is hair length?

on October 10, 2013 at 11:32 PM |Mark RuprightI use the displacement/duration form in my notes and use it in my class. My notes also explain at length how it is related to the more traditional book format. When I work example problems I always use the “change” forms of the kinematic equations to get them used to it. However, when students work on homework they insist on using the book approach.