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Two & Three Dimensional Motion Week 4 -- Lesson 2 Objectives By The End Of This Lecture, You Should Be Able To • - Determine 2-D and 3-D displacements, velocities, and accelerations • - Determine the magnitudes of displacement, velocity, and acceleration vectors • - Combine relative velocities to express the velocity of a particle relative to a coordinate system that is itself moving with respect to anther coordinate system Introduction • Up to this point we have focused on motion that occurred along a single axis. The axis might have been horizontal or vertical, but the motion remained along a single straight line. Most motions do not occur along a single line. Examples of objects that have motions that would not be along a straight line include a baseball that has been struck by a bat, a car as it rounds a curve, a missile that has been fired, the space shuttle after it has been launched, the earth as it orbits the sun and so forth. Displacement In Two Dimensions • We begin by trying to describe the displacement of an object that moves in two dimensions. In the next slide, an object moves from P1 to P2 and then from P2 to P3. We want to know the displacement of the object from P1 at the end of the trip. Adding Vectors Graphically Adding Vectors Graphically Adding Vectors Graphically Adding Vectors Graphically Displacement In Two Dimensions • Remember that when determining the displacement of an object, you are interested in only two locations: the point where the object starts, and the point where the object ends up. With this in mind, we can determine the displacement of the object from its starting position by noting that the path the object actually traced out from P1 to P2 and then to P3 has the same endpoints (and is therefore equivalent for our purposes) as if the object had gone directly from P1 to P3. Displacement In Two Dimensions • In other words, C A B • Notice that each of the terms in the expression has an arrow over them. This is to indicate that the quantity is a vector quantity. Vector algebra is different than ordinary algebra. You must be careful when adding and subtracting vector quantities. Displacement In Two Dimensions • If A has a magnitude of 3 units and B has a magnitude of 4 units, C does not necessarily have a magnitude of 7 units. In fact, C may have any magnitude between 1 unit ( |A-B| ) and 7 units (A+B) depending on how A and B are directed. So, if vector algebra works differently than ordinary algebra, how do you add or subtract vectors? The first step is to break the vectors into parts that lie along common axes. Displacement In Two Dimensions • This process is called breaking a vector into components (or parts of a vector that are oriented at right angles to one another). The next slide shows a three dimensional vector broken into components along the x, y, and z-axes. The vector A can be written as a sum of its three components: Components Of A Vector A A x ˆ A y ˆ Az k i j ˆ Displacement In Two Dimensions • By breaking the vectors A and B into components, each vector has components that lie along the same axes. Components that lie along a common axis add (if they are directed the same way) and subtract (if they are oppositely directed) the same way ordinary numbers do. Adding Vectors By Components Adding Vectors By Components Adding Vectors By Components Adding Vectors By Components C x Ax Bx Adding Vectors By Components C y Ay B y Adding Vectors By Components C C C 2 x 2 y Adding Vectors By Components opposite 1 1 C y adjacent tan C tan x Comment • You need to be very careful when breaking a vector into components and when combining components to form a vector. In chapter 3, the text has some expressions that look like the following: Ax A cos Ay A sin Ay 1 tan A x Comment • And it would appear these expressions are completely general. Then, you read in the paragraph following and find out that they are only valid provided the angle is expressed relative to the positive x-axis. The angles we deal with are not always expressed this way. To be sure you are applying relationships correctly, you should draw the triangles and go back to the definitions of the sin, cos, and tan functions you learned in trig. (review chapter 1 pg 17-18) Vector Analysis of Motion • It will be essential to your physics study to practice the resolution of physical quantities into components. • Try Chapter 3 problems Pg 72-73 1-15 Now. Relative Motion • One of the more straight-forward examples dealing with vectors is that of relative motions that occur at constant velocities. Suppose a person is walking at 1.0 m/s from left to right on a flatbed car that is moving at 10 m/s from right to left relative to the ground. A person on the ground would see: vPC vCG vPC = velocity of person relative to car vCG = velocity of car relative to ground vPG = velocity of person relative to ground v PG v PC vCG Relative Motion Example Continued Notice that the expression shown at the bottom of the last slide is an addition of vectors. The vectors always add, regardless of the directions of the individual vectors involved. The directionality of each vector is taken into account by the vector operation. When the vector expression is reduced to a scalar equation (this can only be done using components), then you add the magnitudes if the vectors point in the same direction and subtract if the vectors point opposite directions: Relative Motion Example Continued vCG vPC vPG vPC vCG m m vPG 1 10 s s m vPG 9 s mˆ v PG 9 i s • Or, the person appears to be heading at 9 m/s to the left when viewed by an observer on the ground. • Two dimensional velocities work the same way, except that you must break the vectors into components before combining them. Give it a try with the problem on the next slide. Try It On Your Own • A plane needs to fly the 675 mile distance from New Orleans to Chicago in 1 hour, 45 minutes in order to keep a flight schedule. There is a 20 mph wind blowing from the Northwest. What constant velocity must the plane have in order to make the trip in the prescribed time? Summary • Two and three dimensional vector quantities are defined analogous to their one dimensional counterparts. • When dealing with vector quantities, break all vectors into components before trying to work with them. Then, if needed, put the parts back together to form a vector. • Be careful to always draw triangles and apply the definitions of sin, cos, and tan when working with vectors to avoid traps and pitfalls. Assignment • Chapter 3 Problems • 1-38 odds pg 45 • Read Lab Activity on Projectiles

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posted: | 3/15/2012 |

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