CH. 2: Kinematics: Describing motion.

[Pages:24]Phys 1110, 2-1 (SJP)

CH. 2: Kinematics: Describing motion.

Our first goal is understanding the motion of objects. The first step is simple: merely DESCRIBING the motion of things. In Ch. 2, we keep life as easy as possible: 1) We'll only talk about "particles": pointlike objects, whose structure is irrelevant. It's an abstraction, a model for real-life objects. (Consider a spherical horse... ) 2) We'll work in one dimension ("1-D"), e.g. a train moving back and forth on a straight track, or a marble tossed straight up and down. (We'll get to more realistic 3-D motion soon enough. The concepts really aren't very different, though)

To describe motion,we need a few basic and critical concepts, quantities, and definitions. We'll use English language words but define them rigorously, mathematically when possible.You'll see that words like "velocity, acceleration, force, energy, momentum..." (which are often sloppy, nebulous, even sometimes synonymous in everyday language!), are, in physics, totally distinct and well defined.

1) POSITION: Where is the object? You need a reference frame to describe position.

A reference frame means a choice of axis and coordinate system: where is the origin,

what units will we use to measure length, which direction will we call positive?

It's a convention, YOU choose the reference frame.

In this coordinate system (labeled arbitrarily by "x"),

x

the object (the black dot) is at position x=+2.5.

0 +1 +2 +3

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Different people can make different choices! If your coordinate system is, say, shifted to the left from mine, the number used to describe the position will then be different, but the position itself will be the same for all observers. Your reference frame may even move! (e.g., a physics lab set up in a train car).

In 1-D horizontal motion, I will usually pick an origin, and let the positive direction be to the right, like in a number line. If I don't state otherwise, you can assume this. But that's a convention. If there's a reason, I can make left be positive. For vertical motion, We sometimes let "up" be +, and sometimes "down". It's important to a draw a coordinate system to define conventions in any problem! Position has a SIGN in 1-D: x=+2.5 and x=-2.5 are totally different positions. (Position has a DIRECTION in more than 1-D. Position is a vector. More on this in Ch. 3)

2) TIME: When does an event occur? You need a reference frame here too: when do you define "t=0" to occur? I label time by "t", which is an INSTANT or POINT in time. E.g. 3:04:25 PM, or "t=2 sec" on a stopwatch.

3a) DISPLACEMENT: This is the net CHANGE in position. E.g. x = +2 m: the object has moved 2 meters to the right (with my usual convention of "+" = "right") The Greek letter there is a "Delta", it always means "change" in this class. x = -2 m means something different, the object has moved 2 meters to the left.

3b) DISTANCE. The total length of the path the object has traveled. It's different from displacement in several ways. It's a positive number, a scalar. If an object moves forward 2 meters and then back 2 meters, the DISTANCE traveled is 4 meters, but the displacement (the NET CHANGE) is zero!

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Example: An object starts at +10, moves right to x=+30, moves back left to +20, then finally moves right to +50, as shown. The overall displacement is "final-initial", x = 50 m - 10m = + 40 m .

start: x1

x2

x3

end: x4

x 0 +10 +20 +30 +40 +50

The overall distance is 20+10+30=60 m (do you see why? Convince yourself now!)

Mathematically: x = x final - xinitial , or if you prefer, xi + x = x f

Position and displacement are useful, but when describing motion, you often care about more, e.g. how fast it's moving (which displacement alone doesn't tell you). We DEFINE a useful measure of "how fast" to be 4a) AVERAGE SPEED = (distance traveled)/(time taken). This is always +, it's called a scalar. In the previous example, if we started at t=0, and then point x2 was reached at 20 sec, point x3 at 30 sec, and the end was at 60 sec, then average speed = (60 m)/(60 sec) = 1 m/s. That's total distance over total time.

4b) AVERAGE VELOCITY = (displacement)/(time taken) This has a sign, or a direction: it's a vector. Given my conventions, v>0 will mean x >0, i.e. the object is moving to the right. If v0, and big (fast!) t

t1

tf

You could ask what the average velocity was, over the whole time interval t=0 to t=tf.

Or, over the shorter interval t=0 to t=t1. (this would be larger, because we were going

faster in those earlier times)

But you might want to know the instantaneous velocity right at the moment t1. That

would be given, graphically, by the average velocity for a

TINY little delta t, right around time t1. On the graph, you x

want the slope as you "zoom in" on the curve at that time. If you try to draw little "chords" (whose slope gives average velocity) and make the two end-points very close

(closer and closer)

x t

together, you will get a better and better approximation to the slope at the point.

t

t1 t1+t

tf

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The instantaneous velocity is, as you can see, the slope of the TANGENT LINE to the

curve at the time t1. Mathematically, this slope is the

5a) INSTANTANEOUS VELOCITY at time t1: v = lim x = dx .

t 0 t

dt

We say v is the "derivative of position with respect to time".

If I use the word "velocity" alone, I generally

x Limit: the tangent line

mean instantaneous velocity. (If I intend "average", I'll explicitly say so)

We may also refer to:

t1 t1+t

x t

t tf

5b) INSTANTANEOUS SPEED (or just speed), defined to be the size of v (the

magnitude of the velocity).

Your speedometer is measuring the size of your instantaneous velocity.

Speed and velocity are closely related, but technically different. (Velocity has a direction, or in 1-D, a SIGN, which is meaningful!) I will try hard not to mix them up, but because they are synonyms in "common usage", I will surely mess up from time to time.

In that last curve, the slope of the tangent line varies from place to place. First, it's quite steep (high velocity), but as time goes by, the tangent gets less and less steep. By the end, (time tf) the curve is horizontal, the tangent line has 0 rise over run, slope=0. Velocity has gone to zero. It makes sense, x isn't increasing with time there, the car is stopped!

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It can be very useful to graph velocity vs. time. It's a graph of the SLOPE of the previous plot, i.e. the derivative of the position vs. time plot.

In that last example, as we just said, it (velocity) starts off large, and decrease towards zero, and would look something like this:

v starts large and +

ends up zero t

t1

tf

If you have a graph (or table) of x versus time, you just measure the slope of the tangent. If you have a formula for x versus time, i.e. the function x(t), then calculus tells us immediately what the velocity is, v = dx/dt. The Appendix A-2 of your text has a lot more details. (See also the appendix at the end of these CH. 2 notes.)

The most common formula we'll use is for the derivative of a simple polynomial: If x = c tn , (where c and n are some constants), then v = dx/dt = (c n) tn - 1. Also, the derivative of a sum is the sum of derivatives.

Example: If x = c*t+b (this is a straight line, it means x is increasing linearly with time),

then v= dx/dt = (c1) t1- 1 + 0 = c .

x

(This makes sense, a straight line has constant slope,

given here by c, of course!)

b

x=ct+b slope=c

t

t1

t2

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