02-1

Physics IClass 02

Two-Dimensional Motion

Rev. 05-Jan-05 GB

02-2

One-Dimensional Motion withConstant Acceleration - Review

B a s i c E q u a t i o n s1 . 00 ttavv 2 . 2

02

1000 )tt(a)tt(vxx

D e r i v e d E q u a t i o n s

3 . )tt)(vv(xx 002

10

4 . 202

100 )tt(a)tt(vxx

5 . 020

2 xxa2vv

02-3

Simple Example of 2D Motion

Throw a ball straight up (1D).

Throw a ball across the room (2D).

02-4

Motion in Two Dimensions

Motion in the X (horizontal) direction isindependent of motion in the Y (vertical) direction.(Neglecting air resistance.)

Strategy: Break the problem into two parts, one forthe X motion and one for the Y motion.

Handle each part like one-dimensional motion.

02-5

Two-Dimensional Equations

B a s i c X E q u a t i o n s1 X . 0xx,0x ttavv 2 X . 2

0x2

10x,00 )tt(a)tt(vxx

B a s i c Y E q u a t i o n s1 Y . 0yy,0y ttavv 2 Y . 2

0y2

10y,00 )tt(a)tt(vyy

02-6

A Special Case of 2D Motion -Projectile Motion

Y

XBy usual convention, we choose X to the right and Yup when we study projectile motion.

T h r o w i n g a b a s e b a l l a n d s h o o t i n g a c a n n o n a r e e x a m p l e s o fp r o j e c t i l e m o t i o n . W e n e g l e c t a i r r e s i s t a n c e a n d a s s u m e t h a t t h eo n l y f o r c e i s g r a v i t y ( d o w n ) .

T h e a c c e l e r a t i o n i n t h e X d i r e c t i o n i s z e r o : 0a x T h e a c c e l e r a t i o n i n t h e Y d i r e c t i o n i s c o n s t a n t a n d d i r e c t e d d o w n :

2

y s/m8.9a

02-7

Projectile Motion -Vx and Vy

02-8

Problem Solving Strategy forProjectile Motion

Make a table, see what you know and what you need to find.

X Yav0

x0 or y0

vf

xf or yf

t-t0 SAME SAME

The common factor in both the X and Y equations is the time atwhich something happens (last row).

02-9

Resolving Vectors into Components

cos so cos

sin so sin

,,

,,

oxoo

xo

oyoo

yo

vvv

v

h

a

vvv

v

h

o

V0

V0,Y

V0,X

Would Vo,y still be related to sine if we were given the other angle, ?

02-10

Example of Projectile Motion -Hit the Falling Target

02-11

Hit the Falling Target Diagram

d

h

The target will drop at the instantthe ball leaves the launcher.

The objective is to adjust the angle so that the ballhits the falling target.

02-12

Hit the Falling TargetTable of Kinematic Quantities

X ball Y ball Y targeta 0 -g -gv0 v0 cos() v0 sin() 0x0 or y0 0 0 hvf v0 cos() (don’t care) (don’t care)xf or yf d ? Same as ball.t-t0 ? SAME SAME

We have all data in the “X ball” column except time.Solve for that first (algebraically).

02-13

Hit the Falling TargetSolving for Time

)cos(vd

)tt()tt()cos(vxx0

0000f

X b a l l Y b a l l Y t a r g e ta 0 - g - gv 0 v 0 c o s ( ) v 0 s i n ( ) 0x 0 o r y 0 0 0 hv f v 0 c o s ( ) ( d o n ’ t c a r e ) ( d o n ’ t c a r e )x f o r y f d ? S a m e a s b a l l .t - t 0 d / [ v 0 c o s ( ) ] S A M E 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 .

02-14

Hit the Falling TargetSolving for Final Y Position

Y ball:

2

00

0f )cos(vd

g21

)cos(vd

)sin(v0y

Y target:

2

0

f )cos(vd

g21

0hy

Setting the two expressions equal, the “g” terms cancel and we are left with

h)cos(

d)sin(

OR

dh

)tan(

02-15

Hit the Falling TargetFinal Solution for Launch Angle

d

h

dh

)tan( means that we aim directly at the target.

Are we “ignoring” gravity? Where did v0 go?

02-16

Class #2Take-Away Concepts

1. X and Y motions are independent.2. In projectile motion problems, the acceleration is

constant = 9.8 m/s2 down (normally -Y direction).3. Strategy for solving projectile motion problems:

Create a table, fill in known quantities, work on findingunknown quantities.

4. Use time to connect information from one column toanother.

02-17

Class #2Problems of the Day

_______1. Two cannon crews, A and B, are practicing shooting theircannons in a large, flat field. Crew A points their cannon at anangle of 30° up from horizontal. Crew B points their cannon 60°up from horizontal. They both fire at the same time and theynotice that their respective cannon balls hit the ground at the sametime. Neglecting air resistance, and assuming both cannon ballsstart from the same height when fired, which cannon ball reachesthe 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.

02-18

Class #2Problems of the Day

2. A pirate ship in a harbor at sea level fired its cannon and hit the top of a hill 310meters above sea level. The cannon ball was in the air for 10.0 seconds. Theelevation angle of the cannon was 30° above horizontal. What was the horizontaldistance from the pirate ship to the top of the hill?Use g = 9.8 m/s2 and ignore air resistance.

h = 310 m

d = ?

= 30°

02-19

Activity #2Projectile Motion

Objectives of the Activity

1. Making sure LoggerPro 3 is installed and workingcorrectly on your laptop.2. Learning how to use LoggerPro for video analysis.3. Using LoggerPro to study projectile motion.

02-20

Class #2 Optional MaterialGalileo

Galileo Galilei (1564-1642)

Galileo studied the motion of free-falling bodies and bodies falling alonginclined planes. He was the first todetermine that the distance traversed bya falling body in equal time intervalsfollows the series 1, 3, 5, 7, 9, …(This is equivalent to saying that thetotal displacement is proportional to thetotal time squared.)

02-21

Aristotle’s Error

Aristotle was unquestionably a genius, but he missed the connectionbetween theory and observation. He taught that a projectile travels ina straight line until it loses the motion imparted to it, then dropsstraight down. Medieval scholars invented the term “impetus” for theimparted motion. A simple observation of a person throwing a rockdisproves this theory, but Aristotle did not think to do that.

Aristotle384-322 B.C.E.

02-22

Parabolic Trajectory

Galileo deduced from his observations that horizontal and vertical motionsare independent. From that he deduced that projectiles travel in curvedpaths and that these curves must be parabolas. He published his results inDiscourses on Two New Sciences, 1638.

Galileo is also responsible for an early form of the Principle of Relativity,which was not revised until Einstein.