Lecture 4: Vectors and Two-Dimensional Motion

Lecture 4: Vectors and Two-Dimensional Motion

Slide 1

·  Last time ... Wraps up our discussion of one-dimensional motion. Discussed concepts of displacement, velocity, and acceleration in 1D, not much more to say, or that is interesting, about motion in 1D.

·  In order to properly discuss mechanics in more than 1D, will need to make extensive use of vectors. And so today, will mostly be focused on a discussion of mathematical properties of vectors, how to manipulate them, and how to apply them to the description of two-dimensional motion.

·  Would strongly urge you to make an attempt to fully master this material on vector properties, as we will be using vectors extensively during the rest of the course, and they are used extensively in PHY 213.

Slide 2

·  All of physical quantities … Recall how we earlier defined difference between vector and scalar quantities …

·  Some examples of scalars are … For example, if we specify a temperature of -40 degrees Fahrenheit, which is pretty cold, well, that number, or information, completely specifies the temperature. When we manipulate scalar quantities, the ordinary rules of arithmetic apply.

·  Now as we have already learned in this course, some examples of vectors are … Now, when it comes to manipulating vector quantities, usual rules of arithmetic concerning addition, subtraction do not apply. We need to learn new rules of vector math.

Slide 3

·  Now as for how we denote vectors … If we write a vector in an equation, we draw an arrow above the variable to indicate that it is a vector. For example, this denotes a vector a, which might appear in the equation: the vector force-F = mass * vector-a.

·  Graphically, represent vectors as arrows … for example, capital-A vector might point off in this direction, and its magnitude is indicated by its length. Capital-B vector is shorter, indicating that its magnitude is less than that of the capital-A vector, and it points off in this direction.

Slide 4

·  First, if we’re going to define new rules for the arithmetic of vectors, let’s start by defining what it means for two vectors to be equal to each other. The rule is: If A and B are two vectors, they are equal if …

·  For example, suppose have four vectors A, B, C, and D here, shown as these red arrows on this x- and y-coordinate system. Note that they are all the same length, and that they all point in the same direction, so according to our definition for equality, all four of these vectors are equal. And so it follows that …

Slide 5

·  Now let’s move on and talk about the rules for vector addition. Now vectors can be added either geometrically or algebraically, we’ll start by discussing the geometric addition of vectors.

·  First, important point, make sure when adding two vectors have the same magnitude. It doesn’t make sense to add a vector with units of m/s to another vector with units of km/hr. Just like when we’re adding scalars, it doesn’t make sense to add a temperature in Fahrenheit to a temperature in Kelvin.

·  To add two vectors geometrically, use so-called “triangle method of addition”. Rules are …

·  So this is illustration of the triangle method of addition. Here, drawn vector A …

Slide 6

·  Now suppose I have two vectors A and B. Does it matter in vector addition if we do A + B or B + A; or does the order of addition matter?

·  So let’s apply our rules of addition … Here have A + B, just as done on previous slide.

·  Now, remembering that we can translate, or move, a vector, and it is still the same vector, let’s change the order of addition. So now, start by drawing B …

·  Find indeed, resultant vector is equal!

·  So, what we say in math/physics language is that, yes, vector addition of A + B = vector addition of B + A, and that vector addition is said to be “commutative”.

Slide 7

·  Now in ordinary arithmetic, of course have negative numbers, -1, -2, -3, -100, etc. When we add -2 to 2, we get 0. Analogously, can define the negative of a vector. But now, since vector has both a magnitude and direction, we have to account for the direction. So we define the negative of a vector to be …

·  This is illustrated in this diagram here. Red indicates some red-vector A. Negative of A, same magnitude, so same length, but opposite direction, pointing 180 degrees opposite direction. So, if were to follow rules for vector addition, if I add A and negative-A, am right back to where I am started. So zero.

Slide 8

·  Now that we’ve defined the negative of a vector, can proceed to define vector subtraction. Subtraction of a vector B from the vector A, A-minus-B, simply defined to be the operation A + negative-B.

·  This is represented graphically as follows. On left, show addition of two vectors A and B, just what we’ve already learned.

·  Now, if I instead want to know the difference, A-minus-B, what I do is add the negative of B, same magnitude, but opposite direction, to A …

Slide 9

·  Final vector arithmetic operations need to define are the multiplication and division of a vector by a scalar …

·  So by multiplying vector by a scalar, we mean vector B = scalar c * vector A …

·  So, for example, where vector B = 2 * vector A, what this means is that vector B is a vector with a magnitude two times that of vector A, but still pointing in the same direction !

Slide 10

·  Finally, want to note that cannot … These definitions do not exist !

Slide 11

·  Suppose that we want to add together more than 2 vectors. The same rules for vector addition still apply. Simply keep adding the tail of each additional vector to the tip of the previous vector.

Slide 12-14

·  So let’s work some examples on the manipulation of vectors …

Slide 15

·  So we’ve already worked through some of the trigonometry required for the addition of vectors … So now let’s give a formal definition to what we mean by the components of a vector.

·  Suppose we have a vector A, this red vector here, shown drawn in this x-y coordinate system.

·  We see that we can express A as the sum of two different vectors: Ax, parallel to the x-axis, and Ay, parallel to the y-axis. These are said to be the “projections” of the vector A along the x- and y-axes: if I “squash” A down onto the x-axis, or “squash” A onto the y-axis.

·  Mathematically, this means I can write the red-vector A as the sum of the vector Ax and the vector Ay. These are called the component vectors of A.

·  The “components” of A then follow from the trigonometry in our problem. We write that the x-component of A, which we denote A_x, is …

·  Written in terms of the components, the magnitude A = … This can just be seen from the Pythagorean theorem. Length, or magnitude of the vector, just the hypotenuse of the triangle,

Slide 16-17

·  Let’s work some examples on manipulating vector components …

Slide 18-19

·  Now that we have learned how to decompose a vector into its components, we can discuss the concept of adding a vector algebraically …

·  So, suppose we want to add this red vector A to the green vector B, by adding the components. First, would begin by decomposing A into its components Ax and Ay, as shown here. Then …

·  Now, if add x-components up, add Ax and Bx, would get a total x-component which is this sum of the red and the green, as indicated here. Now, if add y-components up … So the total vector would be the black vector, which is exactly what we would find if we added A and B graphically, by following the “triangle method of addition”.

Slide 20

·  Let’s work an example …

Slide 21

·  Now you may be wondering: Why have we gone through this long discussion of vectors?

·  The answer is, we will need to use vectors repeatedly in our discussion of motion in 2 dimensions.

·  In 1D-motion … whereas in 2D- (or higher-dimensional) motion …

·  Easy way to see this. Suppose you are in a 1D world, making steps along the x-axis … Now, suppose instead in 2D world, can freely move in 2D. Now, suppose you follow a path like this, first walk in the direction of the red vector, then the green vector, ask the same question …

·  Indeed, displacement from the origin is definitely not equal to the sum of the magnitudes of the steps.

Slide 22

·  So, back to our definition of the displacement … In 2D …

·  To illustrate this graphically, suppose we have our x-y coordinate system, and suppose an object is moving in the x-y coordinate system and follows this curved red path.

·  Suppose at some initial time ti, the position of the object is right here, and so we use the vector ri to denote its position. Then at some later time tf, the position of the object is now here, vector rf. According to our rule, the displacement vector delta-r is just equal to the difference between rf and ri.

·  Now if this vector subtraction is not intuitive, another way to think about this is as follows.

·  At time ti, the object is at position ri, this vector. It then moves, or is displaced, by the magnitude and direction given by the displacement delta-r. Therefore, its final position will be the vector sum of initial vector position, and the displacement!

Slide 23-25

·  Definitions …