Transcript PHYSICS 231 INTRODUCTORY PHYSICS I Lecture 3

PHYSICS 231
INTRODUCTORY PHYSICS I
Lecture 3
Announcement
• HW 2 due:
Wednesday Jan 23
@ 3:59 am
• (MLK Jr. Day on Jan 21)
• Note: related reading for each lecture listed on
Calendar page at PHY 231 website
Main points of last lecture
v f  vi
• Acceleration defined: a 
t
basic equations:
•
Equations with constant
Acceleration:
(x, v0, vf, a, t)
•
1) v  v0  at
1
2) x  (v0  v)t
2
1 2
3) x  v0 t  at
2
1 2
4) x  v f t  at
2
v 2f v 20
5) ax 

2
2
2
a

(g)

9.81
m/s
Acceleration of freefall:
Example 2.9a
A man throws a brick upward from the
top of a building. (Assume the coordinate
system is defined with positive defined
as upward)
B
AA
C
C
At what point is the acceleration zero?
a)
b)
c)
d)
e)
A
B
C
D
None of the above
D
D
E
Example 2.9b
A man throws a brick upward from the
top of a building. (Assume the coordinate
system is defined with positive defined
as upward)
B
AA
C
C
At what point is the velocity zero?
a)
b)
c)
d)
e)
A
B
C
D
None of the above
D
D
E
CHAPTER 3
Two-Dimensional Motion and
Vectors
Scalars and Vectors
• Scalars: Magnitude only
• Examples: time, distance, speed,…
• Vectors: Magnitude and Direction
• Examples: displacement, velocity,
acceleration,…
Vectors in 2 Dimensions
Vector distinguished by
arrow overhead: A
(x, y)
y
Representations:
(x, y)
(r, q)
Cartesian
Polar
x
Vector Addition/Subtraction
•
•
2nd vector begins at end of
first vector
Order doesn’t matter
Vector addition
Vector subtraction
A – B can be interpreted
as A+(-B)
•
Order does matter
Vector Components
Cartesian components are
projections along the xand y-axes
Ax  A cosq
Ay  Asin q
Going backwards,
A
Ax2

Ay2
and q  tan
1
Ay
Ax
Example 3.1a
The magnitude of (A-B) is :
a) <0
b) =0
c) >0
Example 3.1b
The x-component of (A-B) is:
a) <0
b) =0
c) >0
Example 3.1c
The y-component of (A-B) > 0
a) <0
b) =0
c) >0
Example 3.2
Some hikers walk due east from the trail head
for 5 miles. Then the trail turns sharply to
the southwest, and they continue for 2 more
miles until they reach a waterfalls. What is
the magnitude and direction of the
displacement from the start of the trail to the
waterfalls?
5 mi
2 mi
3.85 miles, at -21.5 degrees
2-dim Motion: Velocity
v = r / t
It is a vector
(rate of change of position)
Trajectory
Graphically,
Multiplying/Dividing Vectors by Scalars
• Example:
v = r / t
• Vector multiplied by scalar is a vector: B = 2A
• Magnitude changes proportionately: |B| = 2|A|
• Direction is unchanged:
qB = qA
B
A
2-d Motion with constant acceleration
• X- and Y-motion are independent
• Two separate 1-d problems:
 x, vx, ax
 y, vy, ay
• Connected by time t
• Important special case: Projectile motion
• ax=0
• ay=-g
Projectile Motion
• X-direction: (ax=0)
v x  constant
x  v x t
• Y-direction: (ay=-g) v y, f  v y,0  gt

y  12 (v y,0  v y, f )t
y  v y,0 t  12 gt 2
y  v y, f t  12 gt 2
Note: we ignore
• air resistance
• rotation of earth
gy 
v
2
y, f
2

v
2
y,0
2
Projectile Motion
Acceleration
is constant
Pop and Drop Demo
The Ballistic Cart
Demo
1. Write down x(t)
x  v0,x t
Finding Trajectory, y(x)
2. Write down y(t)
1 2
y  v0,yt  gt
2
3. Invert x(t) to find t(x)
t  x / v0,x
4. Insert t(x) into y(t) to get y(x)
y
v0,y
v0,x
1 g 2
x
x
2
2 v0,x
Trajectory is parabolic
Example 3.3
v0
An airplane drops food to
two starving hunters. The
plane is flying at an altitude
of 100 m and with a velocity
of 40.0 m/s.
How far ahead of the
hunters should the plane
release the food?
181 m
h
X
Example 3.4a
v0
h
q
The Y-component of v at A
a)
b)
c)
D
is :
<0
0
>0
Example 3.4b
v0
h
q
D
The Y-component of v at B is
a) <0
b) 0
c) >0
Example 3.4c
v0
h
q
D
The Y-component of v at C is:
a) <0
b) 0
c) >0
Example 3.4d
v0
h
q
D
The speed is greatest at:
a)
b)
c)
d)
A
B
C
Equal at all points
Example 3.4e
v0
h
q
D
The X-component of v is greatest at:
a)
b)
c)
d)
A
B
C
Equal at all points
Example 3.4f
v0
h
q
D
The magnitude of the acceleration is greatest at:
a)
b)
c)
d)
A
B
C
Equal at all points
Range Formula
• Good for when yf = yi
x  vi,x t
1 2
y  vi,yt  gt  0
2
2vi,y
t
g
x
2vi,x vi,y
g
vi2
x  sin 2q
g
2vi2 cosq sin q

g
Range Formula
vi2
R  sin 2q
g
• Maximum for q=45
Example 3.5a
A softball leaves a bat with an
initial velocity of 31.33 m/s. What
is the maximum distance one could
expect the ball to travel?
100 m
Example 3.6
v0
h
q
D
A cannon hurls a projectile which hits a target located on a
cliff D=500 m away in the horizontal direction. The cannon
is pointed 50 degrees above the horizontal and the muzzle
velocity is 75 m/s. Find the height h of the cliff?
68 m