Motion in 2-dimensions Using Vector Notation

Mark LesmeisterAP Physics C

Dawson High School

© Mark Lesmeister/Pearland ISDSelected graphics from Tipler, “Physics for Scientists and Engineers,”, © 2008 W.H. Freeman and Company.Selected questions taken from Young and Freedman, “Instructor’s Resource DVD for University Physics”, © 2012 Pearson Education Inc.Selected questions taken from Cutnell and Johnson, “Instructor Companion Site for Physics, 9th Edition”, © 2014 John Wiley and Sons

Acknowledgements

KINEMATICS WITH VECTOR NOTATION

Section 1

Position Vectors The position

vector is the vector from the origin to the position of the object.

jiryx

(x,y)r

x

y

Position and Displacement Vectors

The displacement vector is the difference between two position vectors.

12 rrr jir

)()( 1212 yyxx

1r

2r

r

Average Velocity

The average velocity vector is the ratio of the displacement vector to the time interval.

Since the time interval is a scalar, this vector has the same direction as the displacement.

tΔr

vAV

Average Velocity Example

velocity. theof magnitude thefind

Then form.component in stated interval, for this velocity average

theFind . m220 m 40 isposition its , at time later,

minutes Two . m 100 m 400 ofposition a hassailboat A

2 jir

jir

2

1

t

jir

vAV

t

y

t

x

t

jivAV

s 120

m )100220(

s 120

m )40040(

jivAV m/s 0.1 m/s 0.3

Instantaneous velocity

The instantaneous velocity vector is the derivative of the position vector with respect to time.

jir

v

dt

dy

dt

dx

dt

d

jivyx vv

Instantaneous velocity example

vector. velocity ousinstantane theFind

.1800200)( and 2.0100)(by given position a hassailboat A 1 ttyttx

jir

v

dt

dy

dt

dx

dt

d

jiv

218002.0 t

Constant Velocity Model in Two Dimensions

jivr

jiv

tvtvtt

vvvv

yx

yxyx

)()(

.directionsy and x in the elocitiesconstant v theare

and where

Constant Acceleration Model in 2 Dimensions

ravvvv

avv

avrr

00

0

00

2

221

t

tt

Constant Acceleration Model in 2 Dimensions x-direction y-direction

xavv

tavtv

tatvxtx

xxx

xxx

xx

2

)(

)(

20

2

0

221

00

yavv

tavtv

tatvyty

yyy

yyy

yy

2

)(

)(

20

2

0

221

00

RELATIVE VELOCITYSection 2

Relative Velocity A physics student is riding on a light rail

train travelling at 20 mi/hr. The student walks toward the front of the train with a speed of 5 mi/hr in the direction of the train’s motion. How fast does the student appear to be moving relative to the ground?

Relative Velocity Let vAB be the velocity of object A relative to object B. Let vBC be the velocity of object B relative to object C. Then vAC, the velocity of object A relative to object C,

is given by

vAC = vAB + vBC

Relative Velocity Practice

http://cnx.org/content/m42045/latest/?collection=col11406/1.7

PROJECTILE MOTIONSection 3

Projectile motion

The x and y motions are independent of each other. In the x-direction, the motion is

constant velocity. In the y direction, the motion is

constant acceleration, with a =-9.8 m/s2

jir

)()()( tytxt

jiv

)()( tvtv yx

Equations of motion for a projectile x-direction y-direction

0

)(

)(

0

221

00

x

xxx

xx

a

tavtv

tatvxtx

xx

x

vtv

tvxtx

0

00

)(

)(

ga

tavtv

tatvyty

y

yyy

yy

0

221

00

)(

)(

gtvtv

gttvyty

yy

y

0

221

00

)(

)(

©2008 by W.H. Freeman and Company

©2008 by W.H. Freeman and Company

Projectile Motion Example

A physics student throws his cap into the air with an initial velocity of 24.5 m/s at 36.9o from the horizontal. Find the total time the cap is in the air, and the total distance travelled.

©2008 by W.H. Freeman and Company

Projectile Motion Practice

http://cnx.org/content/m42042/latest/?collection=col11406/1.7