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 3 Please report any glitches, bugs or errors to the author: dkoon at stlawu.edu. 3. Vectors and Two-dimensional motion Representation of Vectors Graphical representation of vectors Two-dimensional motion Two-dimensional kinematics page 1 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf
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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 3
Please report any glitches, bugs or errors to the author: dkoon at stlawu.edu.
3. Vectors and Two-dimensional motion Representation of Vectors Graphical representation of vectors Two-dimensional motion Two-dimensional kinematics
FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved
FZX, Chapter 3: VECTORS and TWO-DIMENSIONAL MOTION If we describe the position of an object in two dimensions, we cannot use a simple number line for position. Consider your position on the Earth. If you want to describe your position on the globe unambiguously, you can tell someone your latitude and longitude. Either number alone does not tell your exact location. In another culture, you may find it more convenient to describe your location in different terms (distance and direction from a holy shrine, for instance), but you would still need two numbers, or ‘coordinates’ to describe your two-dimensional position. If you are in an airplane above the ground, you will need three coordinates -- including altitude -- to specify your location. Clearly now, in dealing with two dimensional space, like your location on the Earth, or the position of an object drawn on a blackboard or piece of paper or computer screen, we can no longer use a single number to describe where we are. However, it would still be very convenient to be able to describe the position by a single variable, like we used ‘x’ for position in one-dimension. We call a quantity that needs more than one number to describe it a ‘vector’. To show its difference from nonvector quantities (known as ‘scalars’), we need to write it differently. As of this writing, almost all physics textbooks show vectors by boldfacing the quantity. This is unfortunate, because most lecturers cannot boldface too well in chalk, and students cannot boldface well in pen in their notes. In these notes, I will use an arrow over a quantity’s name [ International symbol for a vector ]
rx to show that it is a vector. This notation is widely used on chalkboards throughout the world (even by authors of texts which use boldfacing in the text!), and will hopefully be adopted by all textbook publishers, eventually. It is vital that you know the distinction between vectors and scalars, so I strongly advise you to do the following exercise: DIY: Take your physics text and turn to the chapter where vectors are introduced. If the text uses boldface to distinguish vectors, go through the chapter and pencil in an arrow above every vector quantity. (If the author uses boldface to emphasize certain words in the text -- horror of horrors! -- do not ‘vectorize’ these.) Why do I ask you to do this? First of all, vectors are a very tough concept to grasp. I will be using arrows on top of symbols on the blackboard to show that the symbol stands for a vector. It is important that you do this too, in order to be sensitized to what quantities are vectors. It does not help that the textbook uses a different (and much less visible) convention to designate vectors. The president of a major publishing house -- I won’t drop names -- tells me that publishers are very conservative, and it is hard to get them to change from what everyone else is using as a convention. (This may also explain why so many intro physics texts are such wan, pallid copies of each other.) Maybe we can do something to get them to change, but in the meantime, you’ve got your pencils. REPRESENTATIONS OF VECTORS: As I mentioned, it is sometimes useful to describe your position in terms of magnitude and direction from some reference point (say Mecca). Other times, it is useful to describe it in terms of x- and y- coordinates (say latitude and longitude). It is sometimes useful to convert between the two. The x- and y-coordinates of a vector are nonvectors or ‘scalars’, and we can designate them as Ax [ x-coordinate of the vector
rA ] rA ] and Ay. [ y-coordinate of the vector
Since these are scalars, they should not be arrowed! If you are confused about this, practice writing the components for the vectors
rB ,
rC , etc., until you think you’ve got it. The magnitude and direction are also scalars, and can be denoted as
A [ magnitude of the vector
rA ] rA ] and θ. [ angle of the direction of vector
‘A’ is just the length of the vector ‘ ‘. The direction is usually measured relative to the positive x-axis and measured in a counterclockwise direction. Draw a vector arrow on a sheet of paper. Now draw a set of x- and y-axes at the TAIL of the vector. The angle you want is the one between the positive x-axis and the vector. This is a useful convention which we will
use when we can. We could have chosen a different direction to be the reference direction. The most important thing is to be consistent in our choice. Now, as for converting between these two alternative ways of describing a single vector -- x- and y-coordinates vs magnitude and direction -- we can use the following sets of equations Ax = A cos θ [Conversion from magnitude and direction to and Ay = A sin θ component form] for converting to ‘coordinate representation’ or A2 = Ax
2 + Ay2 [Conversion from component form to
and tan θ = Ay/Ax magnitude and direction] for going the other way around. Several words of caution are needed here with regard to the last of these equations. Ax and Ay can both be positive, negative, or zero. If Ay and Ax are both negative, Ay/Ax will have the same value it would have if they were both positive and had the same sizes. But the vector represented by two negative coordinates would be pointing in the exact opposite direction as the vector represented by two positive coordinates. When you ask your calculator to take the inverse tangent -- the ‘arctangent’ -- of Ay/Ax, it doesn’t know which of these two vectors you want. So 50% of the time, your calculator will be wrong. These are not good odds to work with. To reduce your chances of being wrong, always draw a picture of the vector, and check to see whether your calculator gives you the right direction. If not, you can fix things by adding 180o to what the calculator gives you. θ = either tan-1 (Ay/Ax) or tan-1 (Ay/Ax) + 180o. Let’s deal with some actual vectors. One of the easiest to work with is displacement. If we walk for 2 hours at 4 miles per hour due North, our displacement is 8mi due North. Unless we know where we started from or where we ended up, we don’t know our position, but we do know our change in position, which is all the displacment is. Let’s make it more interesting. Let’s follow this first displacement with a second displacement, 6mi due East. We can now talk about our ‘net’ displacement, our total displacement, which is the difference between our final position and our initial position, with no concern about where we ended up after the first leg. If we think about it, we are 6mi further East, and 8mi further North than when we started. These are the coordinates of the total vector. What we have seen is if we have two vectors which are ‘added’ (We can say they are added because we talk about the ‘net’ or ‘total’ displacement.), the vector which is their sum has components that are each the sum of the respective components of the two vectors added: If +
rA
rB = , [Vector addition]
rC
then Ax + Bx = Cx AND Ay + By = Cy. We can thus think of the vector equation ‘ +
rA
rB =
rC ‘ as being a sort of shorthand for the two equations below it. This is
why it is important to not forget your arrows: we wouldn’t have known that there was more to the first equation if we hadn’t known it was a vector equation. By the way, there is no similarly simple equation linking the magnitude or the direction of r
to the magnitudes and directions of r
and C ArB . This gives us a reason to prefer to deal with x- and y-components in
many cases, rather than with magnitude and direction. Let’s say I want to take a trip where I end up where I started. If I travel some distance in one direction, and another distance in a different direction, and now want to know where to go to end up back where I started, I can set up the problem as
r r r rA B C+ + = 0r r r
or , C A= − − B where and
rA
rB represent the first two displacements, and
rC is the unknown final displacement. To subtract two
vectors, we simply subtract their x- and y-components separately.
FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved
GRAPHICAL REPRESENTATION OF VECTORS: While it is easy to add two vectors together whose coordinates are known, it is also useful to be able to add them together graphically, just to check that the answer that we get makes sense. This is especially true since there is always that amibguity about the direction of a vector if you use that ‘tan θ = Ay/Ax’ expression to calculate it. Consider how we added two vectors in the first displacement example above. We travelled 8mi North and 6mi East. To draw this, we start at the origin of a system of x- and y-coordinates, and drew a line from the origin to a point 8mi in the positive y-axis. To label this as a vector, we let the ‘tail’ be at the origin and drew the arrow ‘point’ or ‘head’ at the y=+8mi point. This was the starting point for the second leg of the journey. So next we draw a vector going 6mi to the East, which has its ‘tail’ at the ‘head’ of the first arrow. This is how you draw vector addition: Start the first vector at the origin, draw the next arrow starting at the endpoint of the first. The sum of the two arrows is an arrow that starts at the origin and goes to the endpoint of the SECOND arrow. Draw this example now, just to make sure you understand it. To subtract two vectors, -
rA
rB , just add +(-
rA
rB ). That is, invert
rB , by drawing a vector equal but opposite in direction
from rB . This vector, -
rB , must be added to
r, so draw it with its tail at the head of
r -- that is, redraw -A A
rB , starting
where r
ends off. (Remember: I said the magnitude and direction define a vector, but NOT the starting point.) The difference vector has its tail where
r begins, and its head (or arrow) where -
AA
rB ends.
What will you be expected to do with vectors? First of all, you will be expected to be able to convert between the two different representations -- x- and y-coordinate vs magnitude and direction. Secondly, you will be expected to be able to add and subtract vectors, which will include converting between representations if you are given the magnitudes and directions of the original vectors, or are asked for the magnitude and direction of the final vector. Third, you will be expected to sketch vectors graphically. Since this is a much less accurate way of calculating sums and differences, this will mostly be used for checking your algebraic results which come from adding or subtracting coordinates.
FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved
FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved
TWO-DIMENSIONAL MOTION We are in the midst of our first example of how we do physics. We are studying kinematics -- the study of how things move -- and, rather than trying to solve for the general case of how, say, a punted football travels through space, we contented ourselves first with how an object travelling in a straight line moves, consoling ourselves with the thought that we would go to a description of the more general problem once we had understood this simple motion. In fact, we didn’t even study the most general example of ‘one-dimensional’ motion: we contented ourselves with describing motion in which the acceleration is constant, and before we got to that, we contented ourselves with the case in which velocity was constant. In this chapter, we will see that two-dimensional motion can be described using the tools we developed for looking at one-dimensional motion. To give away the secret of this section: we can break any two-dimensional motion into two one-dimensional problems, and consider the motion of the object in the x-axis as if it were independent of the motion in the y-axis. This suggests that the three-dimensional case differs from the one- and two-dimensional cases only in the amount of grunt calculation you have to do, and because the three-dimensional case is rarely needed to describe most motion, we will not even consider it in this course. TWO-DIMENSIONAL KINEMATICS To describe position and how it changes, we note first that position is a vector. Unless we are dealing with something which is constrained to move along a straight path, we need two or three numbers to describe the position. For considering the motion of an object in the xy-plane -- like a blackboard, paper, or a computer screen -- , it is most useful to represent the quantities of motion in terms of x- and y-coordinates, even though we can also use the magnitude and direction to describe the position. First we define the vector position: [ position ]
rx ‘ ‘ is a vector which has components of ‘x’ and ‘y’. Notice the difference between ‘x’ and ‘ ‘. One is a component of the other. They cannot be used interchangeably. If you leave off the arrow above ‘
rx rxrx ‘, it no longer means what you want it to.
There are parallels between proper vector usage and proper English spelling. If you leave out the apostrophe in the word "it’s", you get "its". "It’s" is a contraction for "it is"; "its" means that which belongs to "it". In conversation, you must rely on context to distinguish the two (or you can ask the speaker); in written form, you have to rely on the writer writing it down right. (The two words ‘right’ and ‘write’ are another example, but I’ve digressed enough.) If we change our position, the difference in position is called the displacement: ∆ = - o [ displacement ]
rx rx rx Displacement has coordinates ∆x and ∆y. The velocity is defined as before in terms of the displacement per unit time:
rr
v xtavg =
ΔΔ
[ average velocity ]
[ instantaneous velocity ] r rv vt avg=→
limΔ 0
It has coordinates vx =(∆x/∆t) and vy =(∆y/∆t). We write the coordinates of the initial velocity as either vxo and vyo or vox and voy. The acceleration is defined as before: r
r
a vtavg =
ΔΔ
[ average acceleration ]
r ra at avg=→
limΔ 0
[instantaneous acceleration]
and has components ax =(∆vx/∆t) and ay=(∆vy/∆t). The four kinematics equations still hold in the x-direction and in the y-direction, so we need to write each set of four equations down twice, and to use subscripts on the velocity and acceleration: x = xo + <vx>t y = yo + <vy>t vx = vox + <ax>t vy = voy + <ay>t x = xo + voxt + ½axt2 y = yo + voyt + ½ayt2
FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved
vx2 = vox
2 + 2 ax (x-xo) vy2 = voy
2 + 2 ay (y-yo) [Kinematics equations in two dimensions] I have used brackets to denote averages: <v> = vavg. In the special (but not unusual) case of an object in free fall -- which only occurs if the only thing touching that object is air and we can ignore air resistance (drag) -- we can simplify these expressions. In the free-fall case, there is no acceleration in the x-direction, and the y-acceleration is just ‘-g’. So, we can write... x = xo + vx t y = yo + <vy>t vx = vox v = voy - g t x = xo + voxt y = yo + voyt - ½gt2 vx
2 = vox2 vy
2 = voy2 - 2 g (y-yo)
[FREE-FALL CASE ONLY!] Note: while it is possible that any one of the above formulae will be used in solving a particular problem, the third set of equations is the most commonly useful one. Why? Maybe it is because ‘time’ is the one quantity that connects the x- and y-motion. There is no ‘x-time’ corresponding to, but different from the ‘y-time’. So when in doubt about which equation to use, the old ‘x=xo+vox+½at2’ and its ‘y’ counterpart are usually the best guess.
