Transcript Kinematics - Mr Desantis

What is Kinematics?
Geometry of motion
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.
The Language of Kinematics
The Language of Kinematics
Scalar Quantities:
Quantities that are fully described by magnitude
alone.
Ex:
Temperature = 14 degrees F
Energy =1500 calories
Time = 30 seconds
The Language of Kinematics
Vector Quantities:
Quantities that are fully described by BOTH
a magnitude and a direction.
Ex:
Displacement = 1 mile, Northeast
Velocity = 75 mph, South
Force = 50 pounds, to the right
(East)
The Language of Kinematics
Distance (d): Scalar Quantity
How far an object has traveled during its
time in motion.
Ex: A person walking ½ mile to the end of
the trail and then returning on the same
route: the distance walked is 1 mile.
The Language of Kinematics
Displacement (s): Vector Quantity
A measure of an object’s position measured
from its original position or a reference
point.
Ex: A person walking ½ mile to the end of
the trail and then returning on the same
route: the displacement is 0 miles.
S=0
The Language of Kinematics
Distance: length traveled along a path
between 2 points
End
Start
Displacement: straight line distance
between 2 points
End
Start
The Language of Kinematics
Displacement can be measured as
two components, the x and y
direction:
End
Y displacement
Start
X displacement
The Language of Kinematics
Speed: Scalar Quantity
The rate an object is moving without regard
to direction.
The ratio of the total distance traveled
divided by the time.
Ex: A car traveled 400 miles for 8 hours.
What was its average speed?
Speed = 50 mph
The Language of Kinematics
Velocity (v): Vector Quantity
The rate that an object is changing
position with respect to time.
Average Velocity is the ratio of the total
displacement (s) divided by the time.
The Language of Kinematics
Velocity (v): Vector Quantity
Ex: What would be the average
velocity for a car that traveled 3
miles north in a total of 5 minutes?
The Language of Kinematics
Velocity (v): Vector Quantity
Ex: What would be the average velocity
of a car that traveled 3-miles north and
then returned on the same route
traveling 3-miles south in a total of 22
minutes?
The Language of Kinematics
Acceleration (a): Vector Quantity
The rate at which an object is changing
its velocity with respect to time.
Average Acceleration is the ratio of
change in velocity to elapsed time.
The Language of Kinematics
Acceleration (a): Vector Quantity
Ex: What is the average acceleration of
a car that starts from rest and is
traveling at 50m/s (meters per second)
after 5-seconds?
a = 50m/s – 0m/s
5 sec
a = 10 m/s2
Projectile Motion – Motion in a plane
Motion in 2 directions: Horizontal and
Vertical
Horizontal motion is INDEPENDENT
of vertical motion.
Path is always parabolic in shape and
is called a Trajectory.
Graph of the Trajectory starts at the
origin.
Projectile Motion Assumptions
Curvature of the earth is negligible
and can be ignored, as if the earth
were flat over the horizontal range
of the projectile.
Effects of wind resistance on the
object are negligible and can be
ignored.
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
Horizontal Direction:
• No acceleration: therefore, a = 0.
•
x
Vertical Direction:
• Gravity affects the acceleration. It is
constant and directed downward:
therefore, ay = -g.
Projectile Motion
At the maximum height:
=0
t0
Projectile Motion Formulas
Displacement in general
Professional Development Lesson ID Code: 5009
Projectile Motion Formulas
Horizontal Motion:
The x position is defined as:
Projectile Motion Formulas
Horizontal Motion:
Since the horizontal motion has
constant velocity and the
acceleration in the x direction
equals 0 (ax = 0 because we
neglected air resistance) , the
equation simplifies to:
Projectile Motion Formulas
Vertical Motion:
The y position is defined as:
Projectile Motion Formulas
Vertical Motion:
Since vertical motion is
accelerated due to gravity,
ay = -g, the equation simplifies:
Projectile Motion Formulas
Initial Velocity (vi) 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 Review:
Right triangle:
A triangle with a
90° angle.
Sides:
Hypotenuse, H
Adjacent side, A
Opposite side, O
Opposite
side, O
θ°
Adjacent
side, A
90°
Trigonometric Functions:
Trigonometric Functions:
sin θ° = O / H
cos θ° = A / H
tan θ° = O / A
Projectile Motion Formulas
Projectile Motion Formulas
Projectile Motion Formulas
Horizontal Motion:
Combine the two equations:
and
Projectile Motion Formulas
Vertical Motion:
Combine the two equations:
and
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 = initial velocity = 160 ft/sec
i
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 (ymax).
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?
Answers:
• Maximum height: 21.44 meters
• Total time in the air: 4.18 seconds
• Horizontal Distance: 113.7 meters