Average speed is defined as distance traveled (or driven, if sovcits were around when this book was written neither the author nor I had heard of them yet and the book uses them both) divided by time. Symbolically, ū = d / t. The bar above the u means that it an average - if someone traveled ten feet (are we going to stay in English units throughout the book or switch to metric at some point?) in two seconds, they didn't necessarily at 5 feet/second for the entire two seconds, but that is their average speed.
Now to velocity, the average velocity vector 𝘃̄ which is the displacement vector divided by time (we covered displacement vectors in chapter 1, see "driving 30 miles east".
If we want to measure the speed or velocity at a particular instant, and we're not Zeno, we don't find it impossible to compute it. Since we're not allowed to divide by zero (they taught you that in math class, right? Don't do that, that's where black holes come from) what we do is *almost* divide by zero, we measure the distance traveled over increasingly smaller intervals, and as ∆t (the interval) *approaches* zero in the above equations we approach an instantaneous speed.
If you want to know more about this, you probably should be taking Physics with calculus, but this textbook is for Physics without calculus.
Well now we can define acceleration (and with autocorrect we can even spell it), which happens when an object changes its velocity. It is that change in velocity divided by time, 𝗮̄ = (𝐯 - 𝐯₀) / t, where 𝐯₀ is the velocity at the start, and 𝐯 is the velocity at the end (of time t).
Since the velocities are vectors, so is acceleration. This chapter only deals with motion in a straight line, no curves or turns.
There's a long note about understanding equations instead of just memorizing them and one thing I want to mention is the suggesting there about examining the limiting form of equations, the example they give is that if acceleration is zero, 𝐯 - 𝐯₀ must also be zero, so 𝐯 = 𝐯₀. Yes, that should be obvious, and if the equations give a result that isn't, we have to reconsider whether the fault is with the equation, how we applied it, or with our sense of what is obvious.
(A joke, not in the book: A professor was notorious for leaving complicated demonstrations to the students, with no more than a remark that "It is obvious that..." One day a student interrupted. "Professor, is it really obvious that the second line follows from the first?" The professor looked at the board, wrinkled his brow, paced about the room for a few minutes, then, triumphantly said, "Why yes, it is obvious." In another version he goes back to his office for most of the class and appears breathlessly just before the class ends, with the same conclusion)
OK, now we get into dangerous territory, Galileo's observations about, um, things fall down go boom boom. (there is a hilarious video by a flerf who can't understand how gas gradients work and doesn't believe in gravity, that's something he kept shouting during the video. I think it was at a fellow flerf who was trying not to be quite as ignorant) While Aristotle thought that heavier things do that faster than lighter ones and for about two millennia that was accepted as fact (Aristotle lived 384 - 322 BCE, Galileo 1564 - 1642 CE), Galileo conducted experiments. He increased the time it takes objects to fall by rolling spheres down a slope, we reproduced this experiment in the course where I used this book, and to quote "it is not unlikely that in you laboratory work you will perform an experiment to check his results").
For whatever reason (scientists are baffled!) things fall down at close to 32.2 feet per second squared. Or 9.8 meters per second squared. So here's a sample problem, a stone is dropped from a bridge over water It takes 3 seconds for it to hit the water's self-seeking level So a (don't worry, I'm not calling it g) is 32 ft / sec², v₀ = 0, so the distance is ½at² or 0.5 * 32 * 3 * 3, or 144 seconds (see last equation below)
To sum up, here are all of the equations in scalar form (the same thing happens with vectors but the book wrote them as scalars and I'm tired of going to the site that turns text bold.
s = v̄t (displacement is average velocity multiplied by time, This is the definition of average velocity, rearranged to have displacement on the left side of the equation instead of velocity)
v̄ = ½(v + v₀) (average velocity is the sum of the starting and ending speed divided by two)
v = v₀ + at (final velocity is initial velocity plus acceleration multiplied by time. This is the definition of acceleration, rearranged)
v² = v₀² + 2as (this one is a bit more complicated, we take the definition of average velocity (v̄, two equations above) and plug them into the average velocity equation, the first one in this comment, to get s = ½(v + v₀)t, then replace t by its value from the definition of acceleration and get s = ½(v + v₀)(v + v₀)/a and rearrange that to get v² - v₀² = 2as and then move the v₀² to the right side. Wow, all that algebra we studied turns out to be good for something.
Next time, Forces and Linear Motion
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