Trigonometry and Vectors
Applied to 2D Kinematics
There are two kinds of
quantities…

Vectors are quantities that have both
magnitude and direction, such as…


distance, speed, mass, heat, temperature,
time…etc.
Scalars are quantities that have magnitude
only, such as…

displacement, velocity, acceleration, force…etc.
Direction of Vectors

Vectors are drawn as rays, or
arrows. An angle gives the
direction.

A
x
Typical vector angle ranges
Quadrant II
90o <  < 180o
y
Quadrant I
0 <  < 90o
x
Quadrant III
180o <  < 270o
Quadrant IV
270o <  < 360o
Direction of Vectors


What angle range would this vector have?
What would be the exact angle, and how
would you determine it?
Between 180o and 270o
B


or between
-270o and -180o
x
Magnitude of Vectors



A
The magnitude of a vector is the size of
whatever the vector represents.
Graphically, the magnitude is represented by
the length of the vector.
Symbolically, the magnitude is represented as
│A│or A
If vector A represents a
displacement of three miles to the
north…
Then vector B, which is twice as long,
would represent a displacement of six
miles to the north!
B
Trigonometry needed for
High School Physics

pythagorean theorem
hypotenuse2  adjacent 2  opposite2

opposite

sin, cos, tan (SOHCAHTOA)
sin-1, cos-1, tan-1
θ
adjacent
Graphical Addition of Vectors
B
B
A
R
A+B=R
R is called the resultant vector!
The Equilibrant Vector
B
A
-R
R
A+B=R
The vector -R is called the equilibrant. It is
the inverse of the resultant. Whenever you
add a vector and its inverse, you get zero.
y
Getting
components from
a vector

x component
adjacent
cos 
hypotenuse
Ax
cos 
A
Ax  A cos
A
Ay


Ax
y component
sin  
sin  
opposite
hypotenuse
Ay
A
Ay  A sin 
x
y
Getting a vector
from components

magnitude
hypotenuse2
 adjacent 2  opposite 2
A2  Ax 2  Ay 2
A  Ax 2  Ay 2
A
Ay


Ax
angle
opposite
adjacent
1 opposite
  tan
adjacent
Ay
1
  tan
Ax
tan  
x
Component Addition of Vectors
1)
2)
3)
4)
Resolve each vector into its x- and ycomponents.
Add the x-components together to get
Rx and the y-components to get Ry.
Calculate the magnitude of the
resultant with the Pythagorean
Theorem (R2 = Rx2 + Ry2).
Determine the angle with the inverse
tangent equation ( = tan-1 Ry/Rx.)
Vector Addition Lab
1.
2.
3.
4.
5.
6.
7.
8.
Attach spring scales to force board such that they all
have different readings.
Slip graph paper between scales and board and carefully
trace your set up.
Record readings of all three spring scales.
Detach scales from board and remove graph paper.
On top of your tracing, draw a force diagram by
constructing vectors proportional in length to the scale
readings. Point the vectors in the direction of the forces
they represent. Connect the tails of the vectors to each
other in the center of the drawing.
On a separate sheet of graph paper, add together
graphically. the three vectors
Then add the three vectors by component.
Did you get a resultant? Did you expect one? How big is
it compared to your vectors? What does the resultant
represent?
Relative Motion



Relative motion problems are difficult to
do unless one applies vector addition
concepts.
Let’s consider a swimmer in a river.
Define a vector for a swimmer’s velocity
and another vector for the velocity of the
water relative to the ground. Adding those
two vectors will give you the velocity of the
swimmer relative to the ground.
Relative Motion
Vs
Vw
Vobs = Vs + Vw
Vw
Relative Motion
Vs
Vw
Vobs = Vs + Vw
Vw
Relative Motion
Vw
Vs
Vw
Vobs = Vs + Vw
Practice Problem: You are paddling a canoe in a river that is flowing at 4.0
mph east. You are capable of paddling at 5.0 mph.
a)
If you paddle east, what is your velocity relative to the shore?
b)
c)
If you paddle west, what is your velocity relative to the shore?
You want to paddle straight across the river, from the south to the
north. At what angle to you aim your boat relative to the shore?
Assume east is 0o.
Practice Problem: You are paddling a canoe in a river that is flowing at 4.0
mph east. You are capable of paddling at 5.0 mph.
a)
If you paddle east, what is your velocity relative to the shore?
b)
c)
If you paddle west, what is your velocity relative to the shore?
You want to paddle straight across the river, from the south to the
north. At what angle to you aim your boat relative to the shore?
Assume east is 0o.
Key to Solving 2-D Problems
Resolve all vectors into components
1.


Work the problem as two one-dimensional
problems.
2.

3.
x-component
Y-component
Each dimension can obey different equations
of motion.
Re-combine the results for the two
components at the end of the problem.
Sample problem: A particle passes through the origin with a
speed of 6.2 m/s traveling along the y axis. If the particle
accelerates in the negative x direction at 4.4 m/s2. What are
the x and y positions at 5.0 seconds?
Sample problem: A particle passes through the origin with a
speed of 6.2 m/s traveling along the y axis. If the particle
accelerates in the negative x direction at 4.4 m/s2. What are
the x and y positions at 5.0 seconds?
Projectile Motion
Something is fired, thrown, shot, or
hurled near the earth’s surface.
 Horizontal velocity is constant and
not accelerated.
 Vertical velocity is accelerated.
 In most high school physics classes,
air resistance is ignored.
Launch Angle

Launch angles can range from -90o
(throwing something straight down)
to +90o (throwing something straight
up) and everything in between.
90°
(up)
40°
(down)
-90°
(up and out)
0°
(horizontal)
Sample problem: Playing shortstop, you throw a ball horizontally to the
second baseman with a speed of 22 m/s. The ball is caught by the second
baseman 0.45 s later.
a)
How far were you from the second baseman?
b)
What is the distance of the vertical drop?
Sample problem: Playing shortstop, you throw a ball horizontally to the
second baseman with a speed of 22 m/s. The ball is caught by the second
baseman 0.45 s later.
a)
How far were you from the second baseman?
b)
What is the distance of the vertical drop?
Problem: A soccer ball is kicked with a speed of 9.50 m/s at an
angle of 25o above the horizontal. If the ball lands at the same
level from which is was kicked, how long was it in the air?
Problem: A soccer ball is kicked with a speed of 9.50 m/s at an
angle of 25o above the horizontal. If the ball lands at the same
level from which is was kicked, how long was it in the air?
Projectile over level ground
y
Maximum
Height
Range

x
Mathematically, the trajectory is defined by a highly symmetric
parabola. The projectile spends half its time traveling up, the
other half traveling down.
Acceleration of a projectile
y
g
g
g
g
g
x

Acceleration points down at 9.8 m/s2 for the
entire trajectory of all projectiles.
Velocity of a projectile
y
vx
vy
vx
vy
vx
vy
vx
vx

vy
x
The velocity is tangent to path and can
be resolved into components.
Position graphs for 2-D
projectiles
y
y
x
x
t
t
Velocity graphs for 2-D
projectiles
Vy
Vx
t
t
Acceleration graphs for 2-D
projectiles
ay
ax
t
t

Derive the range equation:
R = vo2sin(2)/g.