Physics I Class 04 Two-Dimensional Motion 04-1 One-Dimensional Motion with Constant Acceleration - Review Basic Equations 1. v v 0 a t t 0 2. x x 0 v 0 ( t t 0 ) 2 a ( t t 0 ) 1 2 Derived Equations 1 x x ( v 0 v)(t t 0 ) 3. 0 2 1 2 x x v ( t t ) a ( t t ) 4. 0 0 0 2 5. v v 0 2a x x 0 2 2 04-2 Simple Example of 2D Motion Throw a ball straight up (1D). Throw a ball across the room (2D). 04-3 Motion in Two Dimensions Motion in the X (horizontal) direction is independent of motion in the Y (vertical) direction. (Neglecting air resistance.) Strategy: Break the problem into two parts, one for the X motion and one for the Y motion. Handle each part like one-dimensional motion. 04-4 Two-Dimensional Equations Basic X Equations 1X. v x v 0 , x a x t t 0 2X. x x 0 v 0 , x ( t t 0 ) 2 a x ( t t 0 ) 1 2 Basic Y Equations 1Y. v y v 0 , y a y t t 0 2Y. y y 0 v 0 , y ( t t 0 ) 2 a y ( t t 0 ) 1 2 04-5 A Special Case of 2D Motion Projectile Motion Y X By usual convention, we choose X to the right and Y up when we study projectile motion. Throwing a baseball and shooting a cannon are examples of projectile motion. We neglect air resistance and assume that the only force is gravity (down). The acceleration in the X direction is zero: a x 0 The acceleration in the Y direction is constant and directed down: a y 9 .8 m / s 2 04-6 Projectile Motion Vx and Vy 04-7 Problem Solving Strategy for Projectile Motion Make a table, see what you know and what you need to find. a v0 x0 or y0 vf xf or yf t-t0 X Y SAME SAME The common factor in both the X and Y equations is the time at which something happens (last row). 04-8 Resolving Vectors into Components o vo, y sin h vo a vo, x cos h vo so so vo, y vo sin vo, x vo cos V0 V0,Y V0,X Would Vo,y still be related to sine if we were given the other angle, ? 04-9 Example of Projectile Motion Hit the Falling Target 04-10 Hit the Falling Target Diagram The target will drop at the instant the ball leaves the launcher. h d The objective is to adjust the angle so that the ball hits the falling target. 04-11 Hit the Falling Target Table of Kinematic Quantities a v0 x0 or y0 vf xf or yf t-t0 X ball 0 v0 cos() 0 v0 cos() d ? Y ball -g v0 sin() 0 ? ? SAME Y target -g 0 h ? Same as ball. SAME We have all data in the “X ball” column except time. Solve for that first. 04-12 Hit the Falling Target Solving for Time d x f x 0 v 0 cos( ) ( t t 0 ) ( t t 0 ) v 0 cos( ) We don’t care what vf is for the ball or target (Y). a v0 x0 or y0 vf xf or yf t-t0 X ball 0 v0 cos() 0 v0 cos() d d/[v0 cos()] Y ball -g v0 sin() 0 (don’t care) ? SAME Y target -g 0 h (don’t care) Same as ball. SAME Next, use the kinematics equation to find yf. 04-13 Hit the Falling Target Solving for Final Y Position Y ball: Y target: d 1 d y f 0 v 0 sin( ) g v 0 cos( ) 2 v 0 cos( ) 1 d yf h 0 g 2 v 0 cos( ) 2 2 Setting the two expressions equal, the “g” terms cancel and we are left with d h sin( ) h OR tan( ) cos( ) d 04-14 Hit the Falling Target Final Solution for Launch Angle h d h tan( ) means that we aim directly at the target. d Are we “ignoring” gravity? Where did v0 go? 04-15 Class #4 Take-Away Concepts 1. 2. 3. 4. X and Y motions are independent. In projectile motion problems, the acceleration is constant = 9.8 m/s2 down (normally -Y direction). Strategy for solving projectile motion problems: Create a table, fill in known quantities, work on finding unknown quantities. Use time to connect information from one column to another. 04-16 Activity #4 Projectile Motion Objectives of the Activity 1. Making sure VideoPoint is installed and working correctly on your laptop. 2. Learning how to use VideoPoint. 3. Using VideoPoint to study projectile motion. 04-17 Class #4 Optional Material Galileo Galileo Galilei (1564-1642) Galileo studied the motion of freefalling bodies and bodies falling along inclined planes. He was the first to determine that the distance traversed by a falling body in equal time intervals follows the series 1, 3, 5, 7, 9, … (This is equivalent to saying that the total displacement is proportional to the total time squared.) 04-18 Aristotle’s Error Aristotle 384-322 B.C.E. Aristotle was unquestionably a genius, but he missed the connection between theory and observation. He taught that a projectile travels in a straight line until it loses the motion imparted to it, then drops straight down. Medieval scholars invented the term “impetus” for the imparted motion. A simple observation of a person throwing a rock disproves this theory, but Aristotle did not think to do that. 04-19 Parabolic Trajectory Galileo deduced from his observations that horizontal and vertical motions are independent. From that he deduced that projectiles travel in curved paths and that these curves must be parabolas. He published his results in Discourses on Two New Sciences, 1638. Galileo is also responsible for an early form of the Principle of Relativity, which was not revised until Einstein. 04-20