fzx physics lecture notes reserved fzx: personal lecture...

FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved

FZX: Personal Lecture Notes from Daniel W. Koon

St. Lawrence University Physics Department

CHAPTER 2

Please report any glitches, bugs or errors to the author: dkoon at stlawu.edu.

2. One-Dimensional Kinematics Free-Fall Graphical Interpretation

page 1 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf


FZX, Chapter 2: ONE-DIMENSIONAL KINEMATICS Here we shall start ‘mechanics’, the study of motion. This study can be divided into ‘kinematics’ -- the study of HOW things move -- and ‘dynamics’ -- the study of WHY things move. While a knowledge of how things move is important and allows us to tell when a solar eclipse several centuries from now will occur to very high precision, it is dynamics that gives physics its real power. By comparing the motion of the planets, for example, to what one would expect, given all their sizes, scientists were able to predict and locate the orbits of Neptune (1846) and Pluto (1930) before ever seeing them. We will approach mechanics in the same way we will approach all of physics. We identify something we wish to study. We consider the simplest model (or description) which describes it. Once we can describe the thing using this model, then we consider slightly more complex models until we have a sufficiently accurate model of what it is we are studying. Physicists are always trying to simplify things in order to explain them. There any number of jokes whose punchlines have the physicist saying something like ‘consider a spherical cow.’ (There are some questions for which the shape of a cow is irrelevant to answer.) With mechanics, we will start with kinematics. Sometimes it is enough to describe what’s going on without trying to explain why it’s happening. We will start with the simplest kind of motion -- motion with uniform speed in a straight line. Then we will consider what happens if the speed changes. In following chapters we will consider motion in two dimensions and circular motion. We will not consider all possible types of motion (or else we would never get to any physics beyond mechanics!), but we will consider enough types of complicating factors that we can be confident of describing most motion that we meet up with. But first, let’s look at simple uniform speed in a straight line. There are two ways of looking at a runner’s progress around a track. One -- the runner’s -- is to say that the runner ran, say 5 miles. The other point of view is that the runner may have run 5 miles, but she ended up where she began, so her change in position is zero. We are going to take the second viewpoint. In discussing motion, we will be interested in the ‘displacement’ or the change in position. Let’s use ‘x’ to describe the position. Then we can describe the displacement as ∆x = x - xo [Displacement] where ‘x’ is the position at some time, and ‘xo’ is the initial position, the position at some time that we will conveniently call t=0. The metric (or ‘SI’ or ‘MKS’) units of position, distance, and displacement are meters. meter [m] [Units of position, displacement] If the speed in the x-direction is uniform and in a constant direction, the position, when graphed versus time, will be a straight line with a nonzero slope. We can measure the rate of change of position by calculating the slope of this graph, which we do by making a right triangle, one side of which is the line, the other two sides of which are lines parallel to the x- and y-axes. We shall call this rate of change the ‘velocity’, v x

t=ΔΔ

= ‘rise’/’run’,

[Constant velocity from ‘x’ vs ‘t’ graph] which is measured, in MKS units, in units of m/s . [units of velocity] Velocity can be either positive or negative or zero. The difference between a velocity of 2m/s and -2m/s is merely the direction. So velocity has both magnitude and direction. We call such a quantity a ‘vector’. We’ll talk more about vectors in the next chapter. If we are interested in just the magnitude of the velocity, we can talk about the ‘speed’, which is 2m/s in either case. Think hard about the difference between velocity and speed: it is an important one that you need to memorize. speed = magnitude of velocity [Speed]


http://en.wikipedia.org/wiki/Spherical_cow


We talked above about a case in which someone or something travels with a constant velocity. If the velocity is not constant, we can still describe the motion over some period of time by giving the ‘average velocity’. Again, we can plot position vs time, consider the position at two different times, and write v x

tx xt tavg = =−−

ΔΔ

0

0

, [Average velocity]

Rearranging things, we get that x = xo + vavgt. [ First law of kinematics ] We use these two formulae all the time. We often tell people the ‘distance’ to the next town or big city in minutes or hours. We assume that the other person will travel with the same average speed as we do, and that the distance is such that, travelling at that speed, it will take 11 minutes or 2 hours or whatever, to get there. Of course, we don’t maintain a constant speed. We must slow down for curves, hills (if our car has a wimpy engine), and rest stops. Still, it is important that you and the other person agree, roughly, on the average speed in order for this to be a useful way of measuring distance.



We can measure the average velocity (and/or speed) if we know the position at two separate times. However, the police will probably pull you over for speeding not if you average a speed above the posted limit, but if your speed exceeds the limit the ‘instant’ that they point the radar gun at you. The velocity you have at some instant is called the ‘instantaneous velocity’. We can find this value by looking at your position vs time on smaller and smaller time intervals, until we reach a time interval during which your speed is hardly changing at all. Then if we measure ∆x and ∆t, that which we measure for vavg won’t be any different than if we took an even shorter time interval. We say that we take the limit of shorter and shorter measurement periods (the limit as ∆t goes to zero) of the average velocity: v v

t avg=→

limΔ 0

=→

limΔ

ΔΔt

xt0

[Instantaneous velocity]

Notice that we write the instantaneous velocity as ‘v’ with no subscripts. If we see no subscripts, we can assume that we are referring to the value of the velocity at some particular instant of time -- the ‘instantaneous’ velocity. What exactly do we mean by this ‘limit’? Draw a circle on a piece of graph paper or lined paper. Call the vertical axis ‘x’ and the horizontal one ‘t’. Now erase the bottom half of the circle. Erase the lefthand side of the semicircle that was left. You now have a graph of position vs time for some imaginary object. Draw in t- and x-axes and give the vertical axis units of ‘meters’ and the horizontal axis units of ‘seconds’. You can calculate the average velocity for this entire curve. Simply measure the change in position, x, from the beginning to the end of the curve, and the change in time, t, and divide these two



quantities. This number is also the slope of the line that connects these two points. What happens if we choose a smaller time interval? The line connecting the two points will be more horizontal, if we start our time interval from the same place. As we make the interval smaller and smaller, the slope will get closer and closer to zero, for this particular example. The instantaneous velocity is zero for the initial time -- v=0 at t=0. A similar approach is to take increasingly powerful magnifying glasses to your graph of position vs time. As you increase the magnification, the curve will look more and more like a straight (but not necessarily horizontal) line. The slope of that straight line is the instantaneous velocity at the time where you are focusing the magnifying glass. Hopefully you have seen from this example that the graphical interpretation of instantaneous velocity is the slope of the ‘tangent’ to a curve of position (‘x’) versus time (‘t’). The tangent to a curve is a line that touches the curve at that point. If you were to look at the curve with a powerful magnifying glass, with a magnification large enough to make the curve look straight, the tangent is the straight line that follows that curve through the region that looks straight under the magnifying glass.

If velocity is changing, then we can treat velocity the same way we treated position when we considered how it changes. We introduce the concept of ‘acceleration’ to describe the rate at which velocity changes: a v

tv vt tavg = =−−

ΔΔ

0

0

[Average acceleration ]

or v = vo + aavgt. [Second Law of kinematics ]



Just as there is ‘average’ and ‘instantaneous’ velocity, so, too, we can have average acceleration and instantaneous acceleration. a a v

tt avg t= =

→ →lim limΔ Δ

ΔΔ0 0

[Instantaneous acceleration]

What is acceleration? If you are following a car on a two-lane highway and you finally get a chance to pass that car, you need to go faster than that car in order to get in front of it. But you have been following this slowpoke for minutes, so that your speed is very close to that driver’s. (Otherwise you would have slammed into that driver’s trunk by now.) To increase your speed in order to pass, you put your foot on the ... ‘accelerator’ -- the gas pedal. Once you’ve passed you see a red traffic light ahead. In order to reduce your speed, you put your foot on the brake. This gives you a negative acceleration, what we usually call a ‘deceleration’. Notice, for the deceleration, that v (vo, so the change in velocity, ∆v = v - vo is negative. (Hence, deceleration is negative acceleration.) This is an introductory algebra-based physics course. We don’t have time to look at physics in too much detail. We could look at what happens when the acceleration is also changing, but for our purposes, motion with uniform acceleration is plenty challenging. Since we will consider acceleration to be constant in this section, we will drop the ‘avg’ for acceleration for the rest of this section. We shall limit ourselves to what we call ‘uniformly accelerated motion’. If the velocity is varying, then during some time interval, between its initial value, vo, and its final value, v, the average value of velocity (assuming constant acceleration) is just vavg = ½ (vo + v) = vo + ½a t.



If we use this as the average velocity for the expression we had for the position, ‘x’, above, then we get x = xo + vot + ½at2 [Third Law of kinematics ] If we solve either of the first two laws of kinematics for ‘t’, and plug that value into this expression, we get the last last of kinematics: v2 = vo

2 + 2 a (x-xo). [Fourth law of kinematics ] To recap these four important laws, we shall write (assuming that the acceleration is uniform):

x = xo + vavgt. [ First law of kinematics ] v = vo + aavgt. [ Second ] x = xo + vot + ½at2 [ Third ] v2 = vo

2 + 2 a (x-xo). [ Fourth ] If the acceleration is NOT uniform, what do we do? If the acceleration is constant over some time interval, and then changes abruptly to another constant value, then we can solve for the motion during each time interval separately. If the acceleration is varying moment by moment, however, we either have to use a more mathematically sophisticated (and difficult) formalism to describe the motion, or we have to settle for a solution in which we do not know the position and velocity at every point in time. Generally, we will take the latter approach, as you will see (several chapters from now) when we get to talking about energy. FREE-FALL: Any object in free-fall accelerates at the same rate, namely ‘g’, which is equal to 9.8m/s2 or 32ft/s2. g = 9.8m/s2 = 32ft/s2 [Free-fall acceleration] By the way, free-fall simply means that you are experiencing the pull of gravity, and there are no other ‘forces’ tugging at you. If you jump off the ground, you are in free-fall from the moment you leave the ground until the moment you land. Your acceleration will be downward whether your velocity is upward (as you rise) or downward (as you fall). This is consistent with your speed decreasing on the way up and increasing on the way down.



GRAPHICAL INTERPRETATION: We mentioned before that velocity can be thought of graphically as the slope of a curve of position vs time. Likewise, acceleration is the slope of a curve of velocity vs time. It is possible to work the other way around too, to get displacement from a curve of velocity vs time or to get the change in velocity from a sketch of acceleration vs time.

If velocity is constant -- let’s assume positive -- the displacement is just the velocity times the time. If we sketch ‘v’ vs ‘t’, we see that the displacement is thus equal to the area under the velocity curve between the initial and final times. If we replace the constant velocity with a straight but sloped curve, we see that the displacement is now equal to the AVERAGE velocity times the difference in time, which is also the area under the curve. If the velocity goes negative, then for that time during which it is negative, we need to subtract the area below the horizontal axis and above the velocity curve to get the displacement.

Likewise, one can show that the area under a curve of acceleration vs time gives the change in velocity. Bear in mind again that, when the acceleration goes negative, you need to subtract the area between the horizontal axis and the acceleration curve.


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