PH 201

Dr. Cecilia VogelLecture 4

REVIEWConstant acceleration

x vs t, v vs t, v vs xVectors

notationmagnitude and direction

OUTLINE2-D motion with acceleration

Projectilesacceleration of gravity

Circular motionconstant speed with acceleration

Special Case: Projectile MotionObject moving with no acceleration except

that of gravity.falling objectthrown object

This is 2-D motion, so vector equations stand forthe y- motion is

ay = -g = -9.8 m/s2

g is a positive #the x-motion is

ax =

Projectile Motion

i ixx x v t

x ixv v

0xa

2 2 2 ( )y iy iv v g y y

ya g

y iyv v gt

212i iyy y v t gt

x-component y-component

Special Case: How long will it take a thrown object to

reach its max height, h?Given voy = vo*sin(theta)

yi= 0

At max height, vy =0

find t

Note – time to go up and back down is twice that

sin( )oy ov v

tg g

Special Case: How high will thrown object go?voy = vo*sin(theta)

yi= 0

At max height, vy =0, y=h

find h2

2yov

hg

Range Height and time only depend on y-component of initial velocity!!!!! Range = horizontal distance covered while going up and back down to original height

Range depends on both components

Horizontal motion is constant velocity, so R= x = voxt

where t=time up and back down

sin( )2 ovt

g

cos( )oR v t

22 cos( )sin( )ovRg

2 sin(2 )ovR

g

Special Case: Uniform Circular Motion

is the (constant) “angular velocity”positive if CCW

So x and y change sinusoidally

Direction angle changes at a constant rate

t

x-axis

Period of MotionPeriod, T, is the time it takes

time to go

If it goes all the way around once, the angle changes by (rad)

2

2

| |T

If it goes all the way around once, the distance traveled is

2

2

d r

rv

T

| |v r

Special Case: Uniform Circular Motion

An object moves in a circle of constant radius, r, with constant speed, v.

Is the object accelerating?

Consider ways of ID’ing acceleration:physical intuition: force needed, jerk feltmath:

AccelerationWhat is the acceleration of the object at

Consider the average acceleration from just before to just after:

Generally: centripetal acceleration is

i v is

fv is

v is

ao s is

Magnitude of Acceleration

Same r and larger v yields: larger accel

Same v and smaller r yields: larger accel

Same T and larger r yields: larger accel even tho r is larger, v is, too!

2

c

va

r

2

2

4c

ra

T

ExampleWhat is the speed and acceleration of the Earth in orbit? r = 93,000,000 mi = 1.5X1011 mT = 1yr = 3.156X107s

= 30,000 m/s

= 0.0059 m/s2

huge circle –

Summary2-D accelerated motion

Projectile motionconstant horizontal speed gravitational acceleration is verticalpath is part of a parabolaspecial case of dropped object

Uniform circular motionconstant speed, changing directionvelocity tangent to circleacceleration toward center of circle