Lecture 10: Vectors, Projectile
Motion
Please sit in the first six rows
Exam I Average = 83
10
9
8
7
6
5
4
3
2
1
0
A
B
C
D
F
Vector Components
Ax  A cos 
y
Ay  A sin 
A
Ay
θ
Ax
x
Ax  A cos 
y
A cos 
Ay  A sin 
Ay
θ
φ
Ax < 0
x
Adding vectors numerically
C  A B
Cx  Ax  Bx
C y  Ay  By
Also called algebraic addition
What would subtraction look like?
Scalar multiplication?
A
A = 15 m
B = 25 m
300
200
B
Find Ax, Ay, Bx, By
C  A B
Cx , C y , C ,C
Examples
The labeled vectors each have length 4 units. For each vector,
what is the component parallel to the ramp?
The labeled vectors each have length 4 units. For each vector,
what is the component perpendicular to the ramp?
Slide 3-27
The vector has length 4.0 units.
What are the x- and y-components of this
vector?
The vector has length 4.0 units.
x- and y-components ?
2.
0
20%
-3
.5
,-
20%
3.
5
20%
2.
0,
-
20%
-3
.5
,2
.0
20%
2.
0,
3.
5
3.5, 2.0
2.0, 3.5
-3.5, 2.0
2.0, -3.5
-3.5, -2.0
3.
5,
2.
0
1.
2.
3.
4.
5.
Velocity as a vector
d d

v
  , same direction as d  
t  t

dr ( x, y )
dt
dx(t )
vx 
dt
dy (t )
vy 
dt
y
Particle moves
from 1 to 2 in
time Δt. Draw
the velocity
vector.
1
2
x
Acceleration as a vector
v f  vi
v dv
a


t f  ti
t dt
dvx
ax 
,
dt
ay 
dv y
dt
Note that unless object is moving in a
straight line, a is not in the direction of v
The diagram below shows two successive positions of a
particle; it’s a segment of a full motion diagram. Which of the
acceleration vectors best represents the acceleration between


vi and vf?
Slide 3-28
Motion on a Ramp
In the Soapbox Derby, young participants build cars with very lowfriction wheels in which they roll down a hill. Cars racing on the
track at Akron’s Derby Downs, where the national championship is
held, begin on a 55 ft section of the track that is tipped 13° from
the horizontal. What is the maximum possible acceleration of a
car moving down this stretch of track? If a car starts from rest and
accelerates at this rate for the full 55 ft, how fast will it be moving?
Slide 3-30
Projectile Motion
• Projectile launched at an angle θ above the
horizontal with an initial velocity of vi.
y
• vix , viy
vi
θ
x
Projectile motion
x and y motion independent
ax  0
vx (t )  vix
x(t )  xi  vixt
a y  g
v y (t )  viy  g * t
1 2
y (t )  yi  viy  gt
2
Problem 3:28
A ball with a horizontal speed of 1.25 m/s rolls off a
bench 1.00m above the floor.
a) How long will it take the ball to hit the floor?
b) How far from a point directly below the edge of
the bench will the ball hit the floor?
Circular Motion
There is an acceleration
because the velocity is
changing direction.
Slide 3-35
Circular Motion
Old vinyl records are 12" in diameter, and spin at 33⅓ rpm when
played. What’s the acceleration of a point on the edge of the
record?
Two friends are comparing the acceleration of their vehicles. Josh
owns a Ford Mustang, which he clocks as doing 0 to 60 mph in a
time of 5.6 seconds. Josie has a Mini Cooper that she claims is
capable of a higher acceleration. When Josh laughs at her, she
proceeds to drive her car in a tight circle at 13 mph. Which car
experiences a higher acceleration?
Slide 3-36
Homework for Today
Reading: 3: 1-6
Problems: 3: CQ1, CQ3, MC19, MC20, 1, 2, 6, 7, 9, 12
MONDAY
Reading: 3: 7-8
Problems: 3: CQ10, MC18, 13, 14, 16, 18, 21,
23, 26, 28, 30, 33