PHY 101 1nd Edition Lecture 7 Outline of Last Lecture I Free Falling Objects II Problem Solving Strategies III 3 1 Vectors and Properties IV Adding Vectors Outline of Current Lecture V 3 2 Components of a Vector VI Vector Addiction Algebraically VII 3 3 Displacement Velocity Acceleration in 2 Dimensions a Average Velocity b Instantaneous Velocity c Average Acceleration d Instantaneous Acceleration Current Lecture 3 2 Components of a Vector A vector can be decomposed into components o Each component by itself is a vector A two dimensional vector has 2 components o A Ax Ay o 1D Vector 1 components o 3D Vector 3 components The magnitude of the components o Ax Acos o Ay Asin o A is the magnitude of the vector A Using the Pythagorean theorem we have o A Ax2 Ay2 1 2 o The angle can also be calculated o tan Ay Ax o tan 1 Ay Ax o The calculated angle is correct only if the vector lies in the 1st or 4th quadrant These notes represent a detailed interpretation of the professor s lecture GradeBuddy is best used as a supplement to your own notes not as a substitute Example o Superman flies from the top of a building along a path The magnitude of the displacement of superman is 100m o A 30 0 because it is clockwise from the axis Find the x and y components Ax Acos 30 0 Ax 100 0 866 Ax 86 6m Ay Asin 30 0 Ay 100 0 5 Ay 50m Vector Addition Algebraically o Graphically tip to tail method o Vector Addition can be done numerically o Suppose R A B o Then R Rx Ry where Rx Ax Bx and Ry Ay By o The x and y components are added separately o Graphically and algebraic methods should give the same answer Example o A hiker travevls 25 km 45 south of east on the first day then travels 40km 60 north of east on the second day o A Determine the components of the hikers displacement on the first and second day day 1 A day 2 B Draw the coordinate vectors Ax Acos 45 Make sure the angle is negative Ax 25 0 707 17 7 km Ay Asin 45 Ay 25 0 707 17 7km Bx Bcos 60 Bx 40 0 5 20 km By Bsin 60 By 40 0 866 34 6 km Total displacement R Rx Ax Bx Rx 17 7 20 37 7km Ry Ay By Ry 17 7 34 6 16 9km o B what is the magnitude of the hikers total displacement for 2 days R Rx2 Ry2 1 2 R 37 7 2 16 9 2 1 2 41 3 km Direction tan 1 Ry Rx tan 1 16 9 37 7 24 1 3 3 Displacement Velocity Acceleration in 2 Dimensions For 2D motion we must consider explicitly the vector nature of displacement velocity and acceleration Displacement use graphs and tip to tail method to figure out the path of the object as vector quantities Average velocity during time interval t o v r t SI Units m s o direction of the average velocity is the same as r instantaneous velocity o v lim t 0 r t SI unit m s o direction of instantaneous velocity along the line that is tangent to the object s path average acceleration o a v t SI Unit m s2 o direction same as v instantaneous acceleration o a lim t 0 v t Important a nonzero acceleration does not necessarily lead to a change in speed The object may simply change the direction of its velocity without changing its speed
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