Physics Quickie: Consistency of Measure
Sometimes maintaining consistent measure in a physical equation is the most important feature determining the final result. Many people get confused as you start to throw mass, length, and time together to create what are called derived units. For instance, area is a function of length and you achieve area by multiplying two lengths together:
10m * 10m = 100m^2
The unit measures on each side of the equation must remain equal. We have two length units on the left side and so we need two on the right side. How confusing can that be when your term is multiplied, but your unit remains? Well, look at the unit as if it were a variable in algebra. What would happen if you multiplied:
5a * 4a = 20a^2
a = 4
(5)(4) * (4*4) = 20 * 16 = 320
(20)(4^2) = 20 * 16 = 320
You see how our scalar values (called coefficients) can be combined, while the variable can't? It has to maintain consistency until the point where it is replaced by an actual value. In physical equations things like meters and feet have no universal meaning. You don't replace meters with some fixed value, nor do you replace seconds. You can operate a conversion, but then the unit simply changes form, and is still a unit.
With that in mind you can start to explain the distance equation a bit better. The confusing term for most people is in squaring the time within the equation. I was surprised how often this is confused after reading a few quick help guides on the Internet that were simply wrong in the way they discussed the workings of the time squaring term. The final result of the distance equation allows for the squaring the the time term to cancel out the square seconds of the acceleration term. The only unit of measure left after this cancellation is the length unit of the acceleration.
d = 0.5at^2
d = m
t = s
a = m/s^2
??m = (0.5)(9.8m)(??s^2) / s^2
??m = (0.5)(9.8m)(??^2)
When working a physical equation you can disregard the units entirely. They aren't needed for the computation to get a resulting value. What they are important for is consistency checking. By making sure the equation we are using is balanced in terms of units we can ensure the final result is going to carry the unit we are looking for. This not only finds situations where you've grossly mis-matched units that measure different properties, but also when you've mismatched units of different measure. Take the following:
??ft = (0.5)(9.8m)(??s^2) / s^2
When the equation is checked for balance we have feet on one side and meters on the other. When we mix the result in meters, but give it the unit feet, we are coming up with an inaccurate measure. In fact, we'd be over 300% off target. Imagine how badly that could effect your next bungie jumping excursion (and yes there have been gross miscalculations of this type before where units weren't properly balanced and people have died). Didn't we almost lose or actually lose one of our planet exploring satellites to a unit error? I'll have to look that up.
The great thing about consistency of measure is that as long as we keep the units consistent we can mix and match. In fact, I could easily replace the gravitational pull of 9.8m/s^2 with the same value in feet 32.17ft/s^2 (acc?) and get a valid result in feet. With that in mind, if you head back to the bouncing ball sample, you can find yet another short-cut that allows you to get rid of the computations in meters, and allows you to turn them into computations in pixels.
a = 30px/s^2
Pixels are a unit of length and measure, and so you can replace meters or feet with pixels in the acceleration constant. Now all of your resulting calculations will be in screen pixels. Because pixels represent a virtual unit, your conversion between real units and pixels are always going to be variable. The conversion factor can be whatever you'd like. This is unlike meters and feet where the conversion factors between them are fixed values.
Bonus: Can anyone draw a relationship between the distance formula of acceleration and the geometric formulas of a triangle?