01-1

Physics IClass 01

1D Motion

Rev. 04-Jan-07 GB

01-2

Definitions

S c a l a r : A n u m b e r – p o s i t i v e , n e g a t i v e , o r 0 .M a g n i t u d e : A b s o l u t e v a l u e – p o s i t i v e o r 0 .V e c t o r : M a g n i t u d e ( o r l e n g t h ) a n d d i r e c t i o n

i n s p a c e .

T i m e : t ( s c a l a r )P o s i t i o n : x

( v e c t o r )

D i s p l a c e m e n t : 0xxx

T i m e i n t e r v a l : 0ttt

A v e r a g e o r m e a n v e l o c i t y i s d e f i n e d a s f o l l o w s :

tx

tt

xxv

0

0avg

01-3

Definitions (Continued)

I n s t a n t a n e o u s v e l o c i t y o r j u s t “ v e l o c i t y ” :

tdxd

tx

limv0t

E x a m p l e : W h e n y o u t a k e a c a r t r i p , y o u g e t t h em a g n i t u d e o f avgv

b y d i v i d i n g t h e c h a n g e i n t h e

o d o m e t e r ( o r d i s t a n c e ) b y t h e h o u r s y o u d r o v e . Y o ug e t m a n y v a l u e s o f v

d u r i n g t h e t r i p b y c h e c k i n g t h e

s p e e d o m e t e r m o m e n t b y m o m e n t .

I f v

i s c o n s t a n t : vv avg

01-4

Definitions (Continued)

A v e r a g e a c c e l e r a t i o n i s d e f i n e d a s f o l l o w s :

tv

ttvv

a0

0avg

I n s t a n t a n e o u s a c c e l e r a t i o n o r j u s t “ a c c e l e r a t i o n ” :

2

2

0t td

xdtdvd

tv

lima

01-5

Definitions (Continued)

Beware: The English word “acceleration” does nothave the same meaning as the physics word. Inphysics, any change in the velocity vector is anacceleration!

Some Additional Physics I Terms:

Speed: Magnitude of velocity vector.Speed Up: Any time the velocity vector’s

magnitude increases.Slow Down: Any time the velocity vector’s

magnitude decreases.

01-6

Components of Vectors

A n y v e c t o r c a n b e w r i t t e n i n c o m p o n e n t f o r m :

kcjbiaa

w h e r e k,j,i a r e u n i t v e c t o r s i n t h e X , Y , a n d Zd i r e c t i o n s r e s p e c t i v e l y . a , b , c a r e c o m p o n e n t s .( S o m e t i m e s y o u w i l l s e e z,y,x u n i t v e c t o r s . )

T h e c o m p o n e n t s o f a v e c t o r a r e s c a l a r s . T h e y c a n b ep o s i t i v e , n e g a t i v e , o r z e r o .

I n o n e d i m e n s i o n : iaa T h e m a g n i t u d e o f a o n e - d i m e n s i o n a l v e c t o r i s t h ea b s o l u t e v a l u e o f i t s c o m p o n e n t : |a | .I f a i s n e g a t i v e , t h e v e c t o r p o i n t s i n t h e n e g a t i v e Xd i r e c t i o n .

01-7

Velocity and Acceleration

We will start with 1D motion. We will deal with the X componentsof displacement, velocity and acceleration: x, v and a.

“When in doubt, draw a graph of velocity versus time.”

t 0

vslope = a

area = x(displacement)

t 1 t

01-8

Constant Acceleration

slope = a

t

v

t0

v0

F o r t h e s p e c i a l c a s e o f c o n s t a n t a c c e l e r a t i o n , t h eg r a p h o f v v e r s u s t i s a s t r a i g h t l i n e . T h e e q u a t i o n i s

00 ttavv

T h i s i s t h e s a m e e q u a t i o n y o u h a d i n m a t h c l a s s f o r al i n e – ]bxmy[ – b u t w i t h d i f f e r e n t s y m b o l s .

01-9

M a t h F a c t : B e c a u s e v e l o c i t y i s t h e d e r i v a t i v e o fd i s p l a c e m e n t , d i s p l a c e m e n t i s t h e a r e a ( i n t e g r a l )u n d e r t h e g r a p h o f v v e r s u s t .

d i s p l a c e m e n t = a r e a = r e c t a n g l e + t r i a n g l er e c t a n g l e : )tt(vbaseheight 00 t r i a n g l e : baseheight

2

1

)tt()vv( 002

1

)tt()]tt(a[ 002

1

202

1000 )tt(a)tt(v)xx(

202

1000 )tt(a)tt(vxx

Displacement withConstant Acceleration

t

v

t0

v0

Important:Check Section 2-10in the book (pg. 27).

01-10

Class #1Take-Away Concepts

1 D E q u a t i o n s o f M o t i o n f o r C o n s t a n t A c c e l e r a t i o n

B a s i c E q u a t i o n s1 . 0f0f ttavv 2 . 2

0f21

0f00f )tt(a)tt(vxx

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

3 . )tt)(vv(xx 0ff021

0f 4 . 2

0f21

0ff0f )tt(a)tt(vxx ( c o m p a r e w i t h 2 . )

5 . 0f20

2f xxa2vv

01-11

Class #1Problems of the Day

_______1. Which one graph below represents a motion for which it would be incorrect touse equations 1-5 to solve a one-dimensional motion problem – even if youbroke the motion into two time intervals? Note: The respective graphs arestraight line segments and v = velocity, a = acceleration.

A)

v

t B)

v

t

C)

a

t D)

a

t

01-12

Class #1Problems of the Day

2. The Faster and the FuriouserTwo teams of students, one from RPI and one from MIT, agreeto a special kind of drag race. The cars will begin byaccelerating in a straight line at a constant acceleration, but at acertain point (which they separately calculate) they will useengine braking to slow down at a constant acceleration. Therules state that each car must reach the finish line with exactlyzero speed. The RPI car begins at +5.0 m/s2 and slows down at–2.5 m/s2. The MIT car begins at +6.0 m/s2 and slows down at–2.0 m/s2. They both start at the same time. The team whosecar reaches the finish line first wins the other team’s car. Thefinish line is 2000 m from the start. Who wins?

01-13

Activity #1Software Loading / 1D Motion

Objectives of the Activity:

1. Making sure Physics I software is installed andworking correctly on your laptop.2. Understanding the basic operation of the motiondetector, cart, and track. (We will use these a lot.)3. Learning general rules and guidelines that willapply to all Physics I activities.4. Review of 1D motion.

01-14

Optional Materialat the End of the Lecture Notes

At the end of most lecture notes, there willbe a section of extra material. This is for theinterest of students who would like to getsome additional depth from the course.We will not be testing on this material andyou are free to skip it.

01-15

Class #1 Optional MaterialDeriving the Other Equations

Equation 3:

3a. Solve 1 for a:0f

0f

tt

vva

3b. Substitute 3a into 2 and simplify.

Equation 4:4a. Solve 1 for v0: )tt(avv 0ff0 4b. Substitute 4a into 2 and simplify.

Equation 5:5a. Solve 1 for (vf–v0): )tt(avv 0f0f

5b. Solve 3 for (vf+v0):0f

0f0f tt

xx2vv

5c. Multiply 5a by 5b and bring 20v to r.h.s.