## Remembering Motion Let’s start "motion" very slowly (no pun intended). > **You already know about motion.** >Since before you started driving, you began to develop an intuition for magnitudes of speeds and speed arithemetic You do that all the time: if you travel at 50 mph for 2 hours, you will have gone how far? 100 miles, right?

You know a lot about speed and could write a sentence-equation that defines it.

Here it is. Admire it because it's going to evolve: $$\text{speed} = \text{distance traveled divided the time that it took}$$ Let's make our word equation more apparent: $$\text{speed}=\frac{\text{distance traveled}}{\text{time that it took}}.$$ Simple! Speed is a *rate* --- a "something" per time. ```{admonition}   Pens out! :class: warning **Question!** A trip: You're driving along at a constant speed and at the $x=10~$mile marker your friend looks at her watch and notes the time to be 4pm. Then you drive a while and at the 90 mile marker she looks again and reads that the time is 6pm. What was your average speed during that time?

**Glad you asked.** Well, it looks like the distance was 80 miles and the time it took was 2 hours. Here's a tool you can use to calcualte this...and anything else in QS&BB. Let Mr Google help. You know that to find an averate speed you divide the distance traveled by the time. Let's do this a couple of ways and you'll see how useful Google has become. The speed is $80/2$ or $40~$mph and you knew that already.

**Good job!** ``` ### Speed units In a "regular" physics course, faculty will obsess about the units of things. Meters, feet, pounds, Volts, kg m/s...the list is almost endless. I'll not *obsess* but of course we'll use units. They'll not get in the way of the physics concepts---sometimes I'll ignore them and just use numbers (see "not obsess" above). We'll need to do some unit conversion, but I'll give them to you or you can get them from Mr Google. (For example, if you wanted to convert 60 mph into km/s...just type "60 mph into km/s" into the search field and behold.) Having said that, let's obsess about highway signs. ```{aside}
``` What are the units of speed? Our signs say, "Speed Limit, 55." What is that? mph? meters per second? km per hour? stadions (look it up) per fortnight? It's silly to just say “55,” but that's what we do. Hold the phone! (That was a Mr Hauswald expression, by the way.) That’s exactly what our speed limit signs *do* say, or actually, don't say. But you know it's "mph." You even intuit speed units without being told! > **Wait.** What about Canada?

> **Glad you asked.** Good question. A Canadian speed limit sign is a lot more informative. It will say "MAXIMUM 100 km/h" (which is about 62 mph). I give full credit to Canadian signs and take off 5 points for absent units to American signs. **This is cool**: I'm not going to worry about unit conversions here (although in a physics class for science majors, that would be a requirement.) Use this tool and type in "(80 miles)/(2 hours) in mph" and see what you get: Calculate it:
* Now type in "(80 miles)/(2 hours) in m/s" and see what happens. * 80 mph is about 128 kilometers, so be tricky and type in: "128 km/2 hours in mph" and be impressed. * How about: "(128 km)/(2 hours) in furlongs per year"? * Oh, how did I know that 80 miles was about 128 km? Why I typed in "80 miles in km" that's how. Feel free to use this when units are an issue or you want to calculate anything by writing out the formula. To us, motion and its measure—speed—is a simple matter. Our cars and even devices on our wrists readily tell us how far we go, how long it takes us to get there, and the rate at which we do it. We can be penalized for traveling at rates that are...too enthusiastic. Speed, or its more sophisticated word-cousin, velocity, is so familiar to us that we hardly pay any attention to just how fundamental it actually is. Speed is a blend of two even more fundamental concepts of space and time. And we're all about space and time in *Quarks, Spacetime, and the Big Bang*. ### Accelerated Motion You've heard about Galileo and towers and Newton and apples and we'll get into that, but I'll bet you remember that when things fall that they go faster and faster as time elapses, so their speeds are not constant but varying. We call that *acceleration* and it too is a rate: $$\text{acceleration}=\frac{\text{velocity change}}{\text{time that it took}}.$$ Step on the gas and you accelerate --- you increase your speed in every time increment --- which maybe is why the long pedal is called that? ### Forces and Acceleration In our unfortunate US units, forces have units of pounds, accelerations have units of ft per second per second (or miles per hour per hour), and masses have units of --- wait for it --- slugs. In the units of science --- and the whole rest of planet earth --- force is measured in Newtons, N, accelerations are meters per second per second, and masses are in grams or kilograms. These units are a subset of the International System of Units (SI), which are the modern "metric units." (Knock yourself out: [https://en.wikipedia.org/wiki/International_System_of_Units](https://en.wikipedia.org/wiki/International_System_of_Units).) Our units are called, "U.S customary units" or sometimes, "Imperial Units," which are very similar. I'll refer back to US units only when it will be helpful for you to make contact with everyday life. From your own experience, or even in a previous class you probably know that pushing on something makes it go faster. Pedaling at a constant force makes your bike go faster, and holding the pedal at a constant pressure makes your car go faster and faster. In the above language, inspired by Newton, we say that a force is proportional to an acceleration: double the force and you double the acceleration. When ever you hear "proportional" to you can turn it into "equal to" with a constant of proportionality and for our purposes here, that constant is the object's mass, $m$. The relationship is then the second of Newton's "laws": $$F=ma.$$ Hey. Our first equation...which is suitable for a T-shirt. ```{admonition}   Pens out! :class: warning **Question!** If I want my acceleration to be 10 meters per second per second and my mass is 90 kg, then I need to apply a force of how many Newtons?

**Glad you asked.** Well, I'd multiply them: $90\times 10=900~$N.

**Good job!** By the way, 10 m/s/s is just about the force of gravity, so if I were to jump out of a tree, 900 N is the force that the earth would be pulling on me as I accelerate to the ground. And the force that the ground would exert on my feet when I land. It's not the fall that's uncomfortable, it's the landing. ``` By the way, Mr Google knows Newton also. * Type in "90 kg * 10 m/s^2 in Newtons" and see what happens. Calculate it:

* Now type in "90 kg * 10 m/s^2 in pounds" and see that he knows how to do the calculation and convert.