Transcript Slide 1

Vector Components &
Adding Non-perpendicular
Vectors
August 28-29 2014
Vector Components
Any vector can be “resolved” into two
component vectors.
How do we calculate
Ax and Ay ?
Use trig!
A
q
Ax
Ay
cos q =
𝐴𝑥
𝐴
sin q =
𝐴𝑦
𝐴
Ax = A cos q Ay = Asin q
Ax is the horizontal component – or x component -of the vector.
Ay is the vertical component – or the y component
– of the vector.
Example: A plane heads east, while the wind moves a plane
north. As a result, the plane moves with velocity of 34 m/s
A
@ 48°relative to the ground.
Calculate the plane's heading and wind velocity.
What does this mean??
It means we need to find the x-component of the plane’s
resulting velocity (= wind velocity) and the y-component of
the plane’s resulting velocity (= plane’s heading).
v = 34 m/s @ 48° . Find vx and vy
cos 48o =
𝑉𝑥
34 𝑚/𝑠
vx = 34 m/s cos 48° = 23 m/s
v
q
vx
vy
sin
48o
=
𝑉𝑦
34 𝑚/𝑠
vy = 34 m/s sin 48° = 25 m/s
A plane moves with a velocity of 63.5 m/s at 32
degrees South of East.
Calculate the plane's horizontal and vertical
velocity components.
cos(320 )
=
vx = ?
320
63.5 m/s
Vy = ?
𝑣𝑥 =
𝑣𝑥
63.5 𝑚/𝑠
63.5 cos(320 )
sin(320 )
=
𝑣𝑦 = 63.5
𝑚
= 53.9 𝐸
𝑠
𝑣𝑦
63.5 𝑚/𝑠
sin(320 )
𝑚
= 33.6 𝑆
𝑠
Problems for you to try individually

A person walks 450 m @ 120 degrees. Find
the x and y component vectors.

A car accelerates 6 m/s2 at 40 degrees. Find
the x and y component vectors.
Problems for you to try individually


A person walks 450 m @ 120 degrees. Find
the x and y component vectors.
225 m west = Ax
390 m north = Ay
A car accelerates 6 m/s2 at 40 degrees. Find
the x and y component vectors.
4 m east = Ax
4 m North = Ay
A
You can find a vector from its components.
This problem may be written differently, but its
exactly the same type of problem we did during
our first lesson on vectors!
Let:
Fx = 4 N
Just add the components to find the overall
Fy = 3 N . vector!
Find magnitude and direction of the vector.
Fx2+ Fy2 = F2
F
q
Fx
Fy
2
2
F= 4 +3 =5N
q = arc tan (¾) = 370
F  5N @37 0
So far, we can …
• Add two (or more) vectors that occur perpendicularly or
in a line (day 1 of vectors)
• Find the x and y components of a vector (today)
• Determine a vector from its components (today)
Can we add two
(or more) vectors if
they occur at nonright angles?
B
A
Now, with a little
graphical
reasoning and
vector components
… we can!
Adding vectors at any angle
1) We start like we always do … moving vectors head to tail.
2) Next, we draw a resultant (as always).
3) Next step is also same as before
… we use the other sides of a right
triangle to calculate the magnitude
and direction of the resultant (C).
By
C
B
A
Ax
Ay
Cx
Bx
Thedifference?
resultant’s xWe
component
The
just haveistoequal
do
to
the
sum
of
the
x
components
of
a little more reasoning to ‘see’ the the
vectors
we are
right
triangle
andadding.
find its sides.
Cy Cx = Ax + Bx
We
always –y ALWAYS
– find
a
Thecan
resultant’s
component
is equal
right
triangle
from of the
to the
sum ofthat
theisy made
components
combining
components of
vectors wethe
arevector
adding.
the
vectors
C =
A + Bwe are adding.
y
y
y
And we can figure all these out with
trig!
Example:
Two people are lugging a heavy suitcase. One pulls
with a 68N force at 24o; the other pulls with a 32 N
force at 65O. What is the total force on the
suitcase?
F = 32 N @ 65°
2
F1 = 68 N@ 24°
F
F
F
2
2
F1
Solving procedure:
1) Move vectors head to tail.
2) Draw resultant.
3) Draw the x and y components of each vector.
4) Find Fx by adding F1x + F2x
5) Find Fy by adding F1y + F2y
6) Calculate F using Pythagorean theorem
7) Calculate q by using tan-1
Example:
Two people are lugging a heavy suitcase. One pulls
with a 68N force at 24o; the other pulls with a 32 N
force at 65O. What is the total force on the
suitcase?
F = 32 N @ 65°
2
F1 = 68 N@ 24°
F
F
F
2
2
F1
Fx = F1x + F2x = 68 cos240 + 32 cos650 = 75.6 N
Fy = F1y + F2y = 68 sin240 + 32 sin650 = 56.7 N
F  Fx2  Fy2  94.5 N
q = arc tan (56.7/75.6) = 36.90
F  945
. N @ 370
You try these individually


V1 = 35 m/s @ 28 degrees
V2 = 40 m/s @ 60 degrees
Find V = V1 + V2
X1 = 4.8 km @ 140 degrees
X2 = 5.3 km @ 30 degrees
Find X = X1 + X2
HINT: If x or y
components point in
opposite directions, then
subtract the smaller from
the larger!
Closure, HW, Exit Ticket
Closure:
How did what we do today …
… relate to our unit statement?
… demonstrate LP traits?
… relate to TOK?
HW: See handout.
HW Quiz next class, test in 2 classes!
Exit ticket: See handout