Transcript Two-Dimensional Motion and Vectors

CP Vector
Components
Scalars and Vectors


A quantity is something that you
measure.
Scalar quantities have only size, or
amounts.


Ex: mass, temperature, volume, love, etc.
Vector quantities have both a size and a
direction (such as N, S, E or W)

Ex: displacement, velocity, acceleration
Vectors are symbolized with
arrows
A
Y component of A or Ay
A vector has components. If the components
are on the axes they are called rectangular
components. The sum of a vector’s
components equals the vector.
A
X component of A or Ax
A vector and its components are
interchangeable. You can either use the vector or its
components, depending on which is easier.
Usually, using the
components is
easier.
Trig functions are used to calculate vector
components


opposite side
sin  
hypotenuse
SOH
adjacent side
cos 
hypotenuse
CAH
opposite side
tan  
adjacent side
TOA
Finding Vector Components
Consider this triangle…
Hypotenuse
A

Opposite
Side
(also Ay)
Adjacent Side
(also Ax)
A cos 

Be careful, the x component is
not always the cos function.

The y component is not always
the sin function.
A

A sin 
Calculate the x and y components
of the following:
x
sin 25 
13
y
cos 25 
13
o
o
y  13cos25
o
Y = 12 m/s
Y
25o
13 m/s
X
x  13sin 25o
X = 5.5 m/s
Calculate the x and y components
of the following:
y
o
sin18 
20.
y  20sin18
Y = 6.2
o
x
cos18 
20
o
20. m
Y
18o
X

x  20.cos18
X = 19 m
o
Drawing the resultant vector from
components (the parallelogram method)
1. Draw your vectors tail to tail
2. Sketch out a parallelogram
that has sides equal to your
original vectors.
3. Draw in the resultant from the
tails to the opposite corner
(going through the parallelogram
diagonally)
Calculating the resultant vector


To calculate the resultant vector, use
the Pythagorean theorem.
a2 + b2 = c2
The Pythagorean theorem only works
for right triangles. Therefore,
rectangular (x & y) components
must be used.
Describing the angle/direction:

Using components to calculate a
resulting vector
Draw and calculate the resultant
vector:
8.0 N
R2 = (8.0)2 + (3.0)2
R2 = 73
R = 8.5
3.0 N
8.0 N
3.0 N
Calculate the angle using inverse
tangent

Use inverse tangent to calculate the
angle.
 = tan-1(opposite/adjacent)
Calculate the angle
8.0 N
Final answer:
8.5 N at 69o _____
3.0 N
8.0 N
3.0 N 
R2 = (8.0)2 + (3.0)2
R2 = 73
R = 8.5 N
 opposite

  tan 
 adjacent
1  8.0 
  tan  
 3.0 
1
  69
o
Describe the angle using north,
south, east, and west.


N
The direction of the vector used for the
“opposite side” is named first.
The direction of the vector used for the
“adjacent side” is named last
N


W
E
W
North of East
S
N
W
E
East of North
S
North of West

E
S
Calculate the angle
8.0 N
Final answer:
8.5 N at 69o E of S
3.0 N
8.0 N
3.0 N 
R2 = (8.0)2 + (3.0)2
R2 = 73
R = 8.5 N
 opposite

  tan 
 adjacent
1  8.0 
  tan  
 3.0 
1
  69
o
Adding Vectors in Real Life…
Step 1: Draw a Vector Diagram
Find The Sum
of A + B + C
A=10.0 m
B=15 m
30.o
20.o
C=10. 0 m
Adding Vectors in Real Life…
Step 2: Create data table holding x and y
components of each vector and the total x
and y components of the resultant vector.
B=15
A=10.0 m
30.o
20.o
C=10.0 m
X
Y
A
9.397 m 3.420 m
B
- 13.0 m
7.5 m
C
0
-10.0 m
B=15. m
15sin30
X
30.o
15cos30
20.o
X
A=10.0 m
10.0sin20
10.0cos20
C=10.0 m
Step 3: Add the vectors along each
axis to get the total resultant x and y
components.
X
Y
A
9.397 m 3.420 m
B
-13.0 m
7.5 m
C
0
-10.0 m
Total -3.6 m
0.92 m
Remember: When adding you round to the least
amount of decimal places (but don’t round until
the end!)
Step 4: Draw a Vector Diagram
showing only the vector axis sums from
step 3.
0.92 m
3.6 m
I dropped the
negative sign
because the arrow
is pointing in the
negative x direction
Step 5: Use the Pythagorean Theorem
(a2 + b2 = c2) to find the magnitude of
the resultant vector.
R2 =(3.6)2 + (0.92)2
R = 3.7 m
0.92 m
3.6 m
R=4m
Step 6a: Use a trig function (usually
tan) to find the angle.
0.92
tan  
3.6
R

3.6 m
0.92 m
  14.3
10o
Step 6b: Specify both magnitude and
direction of the vector.
R = 4 m @ 10o
North of West

R
3.6
0.92 m
Wow! That’s so
much work!
Example: Add the vectors below.
X
Y
A
23.5 m
-8.55 m
B
-10.0 m
0m
TOTAL
13.5 m
-8.55 m
10.0 m
25cos20o = 23.5
20.o
X
25sin20o = 8.55
25.0 m
Example: Add the vectors below.
16 m,
R2 = 13.52 + 8.552
=16 m
32o south of east.
13.5 m
8.55 m

R
Tan  = 8.55/13.5
 = 32.o