# Definitions - Rensselaer Polytechnic Institute Physics I Class 02 Two-Dimensional Motion Rev. 05-Jan-05 GB 02-1 One-Dimensional Motion with Constant Acceleration - Review B a s ic E q u a tio n s 1 . v v 0 a t 2. x x 0 t0 v 0(t t0) 1 2

a (t t0)2 D e riv e d E q u a tio n s 3. x x 4. x x 5. v 2 1 2 ( v 0 v )( t t 0 ) 0

0 v (t t0) v 02 2 a x x 1 2 0 a (t t0)2 02-2 Simple Example of 2D Motion Throw a ball straight up (1D). Throw a ball across the room (2D).

02-3 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. 02-4 Two-Dimensional Equations B a s ic X E q u a tio n s 1 X . v x v 0 ,x a 2X . x x 0 v 0 ,x B a s ic Y E q u a tio n s 1 Y . v y v 0 ,y a

2Y . y y 0 v 0 ,y x t t 0 (t t0) y 1 2 a x(t t0) 2 1 2

a y(t t0) 2 t t 0 (t t0) 02-5 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. T h ro w in g a b a s e b a ll a n d s h o o tin g a c a n n o n a re e x a m p le s o f p ro je c tile m o tio n . W e n e g le c t a ir re s is ta n c e a n d a s s u m e th a t th e o n ly fo rc e is g r a v ity (d o w n ). T h e a c c e le r a tio n in th e X d ir e c tio n is z e ro : a x 0 T h e a c c e le r a tio n in th e Y d ir e c tio n is c o n s ta n t a n d d ir e c te d d o w n : a y 9 .8 m / s 2

02-6 Projectile Motion Vx and Vy 02-7 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). 02-8 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, ? 02-9 Example of Projectile Motion Hit the Falling Target 02-10 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. 02-11 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 (dont care) ? SAME Y target -g 0 h (dont care) Same as ball. SAME We have all data in the X ball column except time. Solve for that first (algebraically). 02-12 Hit the Falling Target Solving for Time x

f x 0 v 0 cos( a v0 x0 or y0 vf xfor yf t-t0 ) (t t0) (t t0) X b a ll 0 v 0 c o s( ) 0 v 0 c o s( ) d d /[v 0 c o s ( )] d v 0 cos(

Y b a ll -g v 0 s in ( ) 0 (d o n t c a re ) ? S A M E ) Y ta rg e t -g 0 h (d o n t c a re ) S a m e a s b a ll. S A M E N e x t , u s e t h e k i n e m a t i c s e q u a t i o n t o f i n d y f. 02-13 Hit the Falling Target Solving for Final Y Position Y ball: Y target:

d 1 d yf 0 v0 sin() g v0 cos() 2 v0 cos() 2 1 d yf h 0 g 2 v0 cos() 2 Setting the two expressions equal, the g terms cancel and we are left with d h sin() h OR tan() d

cos() 02-14 Hit the Falling Target Final Solution for Launch Angle h d h m e a n s t h a tw e

a i m d i r e c t l y a tt h e t a r g e t . tan(

) d A r e w e i g n o r i n g g r a v i t

y ?W h e r e d i d v g o ? 0 02-15 Class #2 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. 02-16 Class #2 Problems of the Day _______1. Two cannon crews, A and B, are practicing shooting their cannons in a large, flat field. Crew A points their cannon at an angle of 30 up from horizontal. Crew B points their cannon 60 up from horizontal. They both fire at the same time and they notice that their respective cannon balls hit the ground at the same time. Neglecting air resistance, and assuming both cannon balls start from the same height when fired, which cannon ball reaches the greatest maximum height? (Do not assume that the cannons and cannon balls are identical.) A) The cannon ball from crew A goes higher. B) The cannon ball from crew B goes higher.

C) Both cannon balls reach the same maximum height. 02-17 Class #2 Problems of the Day 2. A pirate ship in a harbor at sea level fired its cannon and hit the top of a hill 310 meters above sea level. The cannon ball was in the air for 10.0 seconds. The elevation angle of the cannon was 30 above horizontal. What was the horizontal distance from the pirate ship to the top of the hill? Use g = 9.8 m/s 2 and ignore air resistance. h = 310 m = 30 d=? 02-18 Activity #2 Projectile Motion Objectives of the Activity 1. Making sure LoggerPro 3 is installed and working correctly on your laptop.

2. Learning how to use LoggerPro for video analysis. 3. Using LoggerPro to study projectile motion. 02-19 Class #2 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.) 02-20 Aristotles 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. 02-21 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. 02-22