2009 Physics 2111 Fundamentals of Physics Chapter 3 1 Fundamentals of Physics Chapter 3 Motion in 2 &3 Dimensions 1.Moving in 2 &3 Dimensions 2.Position.

2009 Physics 2111 Fundamentals of Physics Chapter 3 1

Fundamentals of Physics

Chapter 3 Motion in 2 &3 Dimensions

1. Moving in 2 &3 Dimensions

2. Position & Displacement

3. Average & Instantaneous Velocity

4. Average & Instantaneous Acceleration

5. Projectile Motion

6. Projectile Motion Analyzed

7. Uniform Circular Motion

8. Relative Motion

Review & Summary

Questions

Exercises & Problems


Position & Displacement

Position Vector:

A vector from a reference point (aka the origin) to the particle.

krjrirr zyx

kjir

523

The vector gives the position of the green ball.


Position & Displacement

Displacement Vector: 12 rrr

kzzjyyixxr

121212

kzjyixr

Consider a particle moving along a path.


Average & Instantaneous Velocity

Average Velocity = ratio of the displacement to the time interval:

Instantaneous Velocity = derivative of particle position wrt time:

A vector in the same direction as the straight line connecting the starting point to the end point.

A vector that is tangent to the particle’s path at the particle’s position.

t

rvavg

.

dt

rd

t

rv

t

lim0


Velocity Components

dt

dzv

dt

dyv

dt

dxv

kvjvivv

dt

rdv

z

y

x

zyx


Average & Instantaneous Acceleration

Acceleration of a particle need not point along the path of the particle.

Average Acceleration = ratio of the change in velocity to the time interval:

Instantaneous Acceleration = derivative of particle velocity wrt time:

If the velocity of a particle changes in either magnitude or direction, the particle is subject to an acceleration.

t

vaavg

.

kajaiadt

vda zyx


Projectile Motion

The particle is launched with initial velocity vo:

For a projectile, the motions in the vertical and horizontal planes are independent of each other:

horizontal motion with zero acceleration

vertical motion with constant downward acceleration

sin

cos

00

00

000

vv

vv

jvivv

y

x

yx

gay

0xa

2-Dimensional Motion: a projectile moves in a vertical plane subject to the downward acceleration due to gravity.


Projectile Motions

gay

0xajvivv yx

000


Identical Vertical Motion

Two balls are released simultaneously

00 xv

00 xv

The vertical motions are identical.


Projectile Motions

vx = v0x

vy = - v0yjvivv yx

000

g

22

100 tatvxx xx

000 cos

0

vvv

a

xx

x

tvxx 000 cos

22

100 tatvyy yy

000 sinvv

ga

y

y

22

1000 sin tgtvyy

tgvvy sin0

Equations that describe Projectile Motion


Trajectory of a Projectile

Particle Trajectory:

22

100 sin tgtvy

tvx 00 cos

2

00

2

0cos2

tan

v

xgxy parabolic path


Range of a Projectile

0

20 2sin

g

vR

22

1000 sin tgtvyy

00 yy

tvxx 000 cos

RANGE0 xx

cossin22sin


Range equation


Maximum Range of a Projectile

0

20 2sin

g

vR

R is a maximum for 0 = 45o.


Range of a Projectile


Projectile Motion Applet

Launch Internet Explorer Browser.lnk

http://physics.uwstout.edu/physapplets/virginia/www.phys.virginia.edu/classes/109n/more_stuff/applets/ProjectileMotion/enapplet.html





The Ball Hits the Can Every Time!

Magnet (M) releases the can just as the projectile leaves the blow gun (G).

During the time-of-flight, both the projectile and the can fall the same distance, h, under the constant acceleration –g.


Projectiles


Object Falling from a Moving Plane

released horizontally

h = 500m, v0 = 55 m/s

At what value of should the payload be released?

ght

tgh

tgtvy

/2

sin2

21

22

100

00

tvx

tvx

tvx x

0

00

0

cos

h

xtan

tgv

vv

y

x

0


Cannon Firing at Pirate Ship

V0 cannonball = 82 m/sPirate ship is 560 m offshorea) What for cannonball to hit shipb) Safe range for the ship?


Great Zacchini

The “flight of Emanuel Zacchini” over 3 ferris wheels

v0 = 26.5 m/s, 0 = 530, h0 = 3.0 mDoes he clear first Ferris wheel ?If hmax is above second Ferris Wheel, by how much does he clear it?


Uniform Circular Motion

The velocity is changing, but the speed is not.Hence, the particle is accelerating.

The acceleration is always directed radially inward.

Centripetal Acceleration:

r

va

2

v

rTperiod

2

The velocity is always tangent to the circular path.

Let’s prove this.


Circular Motion

r

r

v

v

r

tv

v

v

r

v

t

va

2


“Reference Frame B moves @ constant velocity wrt Reference Frame A”

Relative Motion in One Dimension

Differentiate wrt time

BAPBPA vvv


PBPA aa


Relative Motion in 2 Dimensions

Two observers watching the motion of particle P from the origins of reference frames A and B, while B moves at constant velocity vba relative to A.

Observers in different reference frames that move at constant velocity relative to each other will measure the same acceleration.

BAPBPA vvv

BAPBPA rrr

PBPA aa




Relative Motion

WGPWPG vvv

vpw = 215 km/h, south of east

vwg = 65.0 km/hr 200 E of N

What is vpg and


Example

y – y0 = v0 sin 0 t - ½ g t2

22

128.960sin60200 ttm

smo

sm

081.4060.102 tt

storst 00.36.13

R = x – x0 = v0 cos 0 t

sR os

m 6.1360cos60

mR 408


Example: How fast should she be going?

22

10 tgyy

22

128.95 tm

sm

st 01.1

tvxx 00

sm

s

m

t

xxv 8.23

01.1

2400

hmi

hkmv 53850


Example: Can He Make the Jump?

x – x0 = v0 cos0 t

y – y0 = v0 sin 0 t - ½ g t2

0 = 0

v0

y0

y = 0

He can’t make it!

t = 0.990 s

x – x0 = 4.5 m

2009 Physics 2111 Fundamentals of Physics Chapter 3 1 Fundamentals of Physics Chapter 3 Motion in 2 &3 Dimensions 1.Moving in 2 &3 Dimensions 2.Position.

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