Kinematics
Geometry of Motion
What is Kinematics?
Kinematics
is the study of the geometry
of motion and is used to relate
displacement, velocity, acceleration and
time without reference to the cause of
motion.
Projectile Motion Assumptions
The variations of gravity (g) with
respect to differing altitudes is
negligible and can be ignored.
Gravity is constant:
or
Projectile Motion
First step:
To analyze projectile motion, separate
the two

dimensional motion into
vertical and horizontal components.
Projectile Motion
Horizontal Direction, x
, represents the
range, or distance the
projectile
travels.
Vertical Direction, y,
represents the
altitude, or height, the projectile
reaches.
Projectile Motion Formulas
Initial Velocity (v
i
) can be broken
down into its x and y components:
Projectile Motion Formulas
Going one step further:
There is a right triangle relationship between
the velocity vectors
–
Use Right Triangle
Trigonometry to solve for each of them.
Right triangle:
A triangle with a
90
°
angle.
Right Triangle Review:
Opposite
side,
O
Adjacent
side,
A
θ
°
90
°
Sides:
Hypotenuse,
H
Adjacent side,
A
Opposite side,
O
Trigonometric Functions:
sin
θ
°
=
O
/
H
cos θ
°
=
A
/
H
tan
θ
°
=
O
/
A
Trigonometric Functions:
Projectile Motion Problem
A ball is fired from a device, at a rate
of 160 ft/sec, with an angle of 53
degrees to the ground.
Projectile Motion Problem
•
Find the x and y components of V
i
.
•
At the highest point (the vertex) what
is the altitude (h) and how much
time has elapsed?
•
What is the ball’s range (the distance
traveled horizontally)?
Projectile Motion Problem
Find the x and y components of V
i
.
V
i
= initial velocity = 160 ft/sec
Projectile Motion Problem
Find the x and y components of V
i
.
Projectile Motion Problem
Find the x and y components of V
i
.
Projectile Motion Problem
At the highest point (the vertex), what is
the
altitude (h) and how much time has
elapsed?
Start by solving for time.
Projectile Motion Problem
At the highest point (the vertex), what is the
altitude (h) and how much time has
elapsed?
Now using time, find h (y
max
).
Projectile Motion Problem
What is the ball’s range (the distance
traveled horizontally)?
It takes the ball the same amount of time to reach its
maximum height as it does to fall to the ground, so total time
(t) = 8 sec. Using the formula:
Projectile Motion Problem

2
A golf ball is hit at an angle of 37
degrees above the horizontal with a
speed of 34 m/s. What is its maximum
height, how long is it in the air, and how
far does it travel horizontally before
hitting the ground?
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