Keeping your units straight

The best thing I ever learned in a physics class was one of the very first things: keep your units straight. When doing complex problems that contain multiple steps, there’s nothing more important. The key is writing everything out long-hand.

We all know that Velocity is distance per time. Let’s call it METERS per SECOND. Instead of writing it with a slash m/s, write them above and below a horizontal line:

METERS
————–
SECOND

This is what Velocity is. Meters divided by seconds. By writing units like this, we can make sure our answer is at least in the correct domain (distance, time, acceleration, velocity, etc…) We do this by crossing out units above and below the lines wherever we can.

Molly is flying at 10,000 meters per second. How long will it take her to go 125,000 meters? These problems drive people crazy written out like this. Let’s straighten up the units:

10,000 METERS
———————–
SECONDS

The question is “How long will it take?” so the answer will be in SECONDS. That means, something needs to cancel out those METERS up top. How do we cancel out METERS up top? With METERS on the bottom.

10,000 METERS                               1
———————–             X         ————————
SECONDS                                 125,000 METERS

Wait. Why am I multiplying times a fraction? Because all I care about are the UNITS, not the numbers. This is the new thinking you have to get used to. Remember that any number can be “flipped around” by dividing it into 1. What we are really doing here is DIVIDING by 125,000. But, instead of creating a third level, which is confusing, we’re lining everything up nice and neat.

When I multiply those two, I get:

10,000 METERS
———————————————
125,000 METERS SECONDS

On the bottom, you see METERS SECONDS, which just means they’re multiplied together. But the METERS cancel out, top and bottom, which means the above is really:

10,000
——————————-
125,000 SECONDS

Let’s simplify those numbers up top and bottom by dividing both by the smaller number (that will cancel out the top and turn it into a 1 and give us some number on the bottom):

1
————————
.08 SECONDS

Weird. What kind of answer is that? Wait! We’re not supposed to be looking at the numbers, we’re supposed to be concentrating on the UNITS! Our answer is supposed to be in SECONDS, but the figure above is in 1 / SECONDS. We need to flip that number around. How do we do that? You just divide the bottom number into the top (or use the 1/x button on your calculator). When we do that, the answer is:

12.5 SECONDS. That’s how long it will take Molly to fly 125,000m if she’s flying at 10,000m/s.

Now, the astute reader might say, “I saw that right off! Why do all these steps?”

Well, this is a very simple example to show you how to think in units. Let’s look at a harder question:

Molly weighs 125 pounds. She is standing on the surface of the Earth, where gravity’s acceleration is 9.8 meters per second squared. How much force is she exerting on the Earth?

Holy mackerel. Where to even begin? Let’s start by looking up the formula for “force.” Our book tells us that force is:

F = kg x m / s^2

Looks like gibberish to me. Let’s rewrite it:

FORCE=

KILOGRAMS   METERS
————————————–
SECONDS   SECONDS

Check this out. I might take a few readings to see what I’ve done.

125 POUNDS             1 KILOGRAMS                 9.8 METERS
———————–  x   —————————–   x    —————————
1                                         2.2 POUNDS                 SECONDS SECONDS

We can cancel out the POUNDS and multiple all the way through and get:

1225 KILOGRAMS METERS
———————————————-
2.2 SECONDS SECONDS

Or F = 556.8 kgs m/s ^2

By following our units and canceling them out, we know we haven’t divided when we should have multiplied, or vice versa. We used 1 kg/2.2 pounds to get rid of the pounds and replace them with kilograms and treated that just like any other calculation.

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