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