Physics 103: Lecture 4 Vectors - Motion in Two Dimensions Today’s lecture will be on Vectors Two dimensions projectile motion 9/17/2018 Physics 103, Spring 2004, U.Wisconsin 1

Physics 103, Spring 2004, U.Wisconsin One Dimension } Define origin Define sense of direction Position is a signed number (direction and magnitude) Displacement, velocity, acceleration are also vectors specified just by signed numbers Reference Frame …-4 -3 -2 -1 1 2 3 4… 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Two Dimensions Position can be anywhere in the plane Again, select an origin Draw two mutually perpendicular lines meeting at the origin Select +/- directions for horizontal (x) and vertical (y) axes Any position in the plane is given by two signed numbers A vector points to this position The square of its length is, R2= Rx 2+ Ry 2 The angle of that vector is,  = tan-1(Ry / Rx) R Ry Rx = R cos  Ry = R sin  q Rx 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Preflight 4, Q 1 & 2 Can a vector have a component bigger than its magnitude? Yes No The square of magnitude of a vector is given in terms of its components by R2= Rx 2+ Ry 2 Since the square is always positive the components cannot be larger than the magnitude 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Preflight 4, Q 3 & 4 The sum of the two components of a non-zero 2-D vector is zero. Which of these directions is the vector pointing in? 45o 90o 135o 180o 135o -45o The sum of components is zero implies Rx = - Ry The angle,  = tan-1(Ry / Rx) = tan-1 -1 = 135o = -45o (not unique, ± multiples of 2 ) 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Vector Algebra Analytical method Add the components separately to get the components of sum vector Rx = R1x + R2x Ry = R1y + R2y Scalar multiplication of vector Can change magnitude and sign Multiply all components by scalar Components of sR are sRx and sRy Negation of vector (multiplying by -1) Reverse signs of both components Vector points in opposite direction R=R1+R2 D=R2-R1 y R1 R2 D x 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Vectors! 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Summary of Lecture 3 Free-fall: ay = -g = -9.81 m/s2 Equations with constant acceleration x = v0t + 1/2 at2 v = at v2 = v02 + 2a x Free-fall: ay = -g = -9.81 m/s2 y = y0 + v0yt - 1/2 gt2 vy = v0y - gt vy2 = v0y2 - 2gy 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Two Dimensional Motion 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Range of Soccer Ball Dimensional Analysis: Strategy: 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Kinematics in Two Dimensions x = x0 + v0xt + 1/2 axt2 vx = v0x + axt vx2 = v0x2 + 2ax x y = y0 + v0yt + 1/2 ayt2 vy = v0y + ayt vy2 = v0y2 + 2ay y x and y motions are independent! They share a common time t 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Kinematics for Projectile Motion ax = 0 ay = -g y = y0 + v0yt - 1/2 gt2 vy = v0y - gt vy2 = v0y2 - 2g y x = x0 + vxt vx = v0x x and y motions are independent! They share a common time t 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Projectile Motion: Maximum height reached Time taken for getting there 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Projectile Motion: Maximum Range 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Soccer Ball 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Higher the shell flies, the longer it takes. A battleship simultaneously fires two shells at enemy ships from identical canons. If the shells follow the parabolic trajectories shown, which ship gets hit first? 1. Ship A. 2. Ship B. 3. Both at the same time Higher the shell flies, the longer it takes. What should the captain order if he wants to hit both ships at the same time? A B 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Lecture 4, Pre-Flight 5&6 You and a friend are standing on level ground, each holding identical baseballs. At exactly the same time, and from the same height, you drop your baseball without throwing it while your friend throws her baseball horizontally as hard as she can. Which ball hits the ground first? 1. Your ball 2. Your friends ball 3. They both hit the ground at the same time correct 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Lecture 4, Pre-Flight 5&6 (great answers) They both have the same initial vertical component with the same acceleration due to gravity, therefore they hit the ground at the same time. No matter how much horizontal velocity is put on an object it still falls at the same rate as any other dropped object. y = y0 + voyt - gt2/2 v0y = 0 and y=0 Therefore, t=sqrt(2y0/g) Result is independent of v0x 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Question? Without air resistance, an object dropped from a plane flying at constant speed in a straight line will 1. Quickly lag behind the plane. 2. Remain vertically under the plane. 3. Move ahead of the plane 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Lecture 4, Pre-Flight 7&8 A flatbed railroad car is moving along a track at constant velocity. A passenger at the center of the car throws a ball straight up. Neglecting air resistance, where will the ball land? 1. Forward of the center of the car 2. At the center of the car 3. Backward of the center of the car correct 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Great Answers! The train and the ball have the same horizontal velocity and by throwing the ball straight up, the horizontal component is not changed. The ball has no acceleration in the horizontal direction. Therefore, the balls remains directly above the center of the train at all times during the flight and would fall directly back toward the center of the train. 9/17/2018 Physics 103, Spring 2004, U.Wisconsin

Physics 103, Spring 2004, U.Wisconsin Summary Projectile Motion y = y0 + v0yt - 1/2 gt2 vy = v0y - gt vy2 = v0y2 - 2g y x = x0 + v0t v = v0x 9/17/2018 Physics 103, Spring 2004, U.Wisconsin