What is Kinematics? Geometry of motion
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Transcript of What is Kinematics? Geometry of motion
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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.
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The Language of Kinematics
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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
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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)
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The Language of Kinematics
Distance (d): Scalar QuantityHow 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.
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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
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The Language of Kinematics Distance: length traveled along a path
between 2 points
StartEnd
Displacement: straight line distancebetween 2 points
Start
End
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The Language of Kinematics
Displacement can be measured as two components, the x and y direction:
Start
End
X displacement
Y displacement
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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
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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.
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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?
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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?
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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.
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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 a = 10 m/s2
5 sec
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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.
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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.
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Projectile Motion Assumptions
The variations of gravity (g) with respect to differing altitudes is negligible and can be ignored.
Gravity is constant:
or
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Projectile Motion
First step:
To analyze projectile motion, separate the two-dimensional motion into vertical and horizontal components.
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Projectile Motion
Horizontal Direction, x, represents the range, or distance the projectile travels.
Vertical Direction, y, represents the altitude, or height, the projectile reaches.
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Horizontal Direction: • No acceleration: therefore, ax = 0.•
Vertical Direction: • Gravity affects the acceleration. It is constant and directed downward:
therefore, ay = -g.
Projectile Motion
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Projectile Motion
At the maximum height:
= 0
t0
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Displacement in general
Projectile Motion Formulas
Professional Development Lesson ID Code: 5009
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Projectile Motion Formulas
Horizontal Motion:
The x position is defined as:
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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:
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Projectile Motion Formulas
Vertical Motion:
The y position is defined as:
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Projectile Motion Formulas
Vertical Motion:
Since vertical motion is accelerated due to gravity, ay = -g, the equation simplifies:
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Projectile Motion FormulasInitial Velocity (vi) can be broken down into its x and y components:
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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.
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Right triangle:A triangle with a 90° angle.
Right Triangle Review:
Hypotenuse, HOpposite side, O
Adjacent side, A
θ° 90°Sides:Hypotenuse, HAdjacent side, AOpposite side, O
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Trigonometric Functions:
sin θ° = O / H
cos θ° = A / H
tan θ° = O / A
Trigonometric Functions:
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Projectile Motion Formulas
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Projectile Motion Formulas
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Projectile Motion Formulas
Horizontal Motion:
Combine the two equations:
and
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Projectile Motion Formulas
Vertical Motion:
Combine the two equations:
and
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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.
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Projectile Motion Problem
• Find the x and y components of Vi.
• 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)?
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Projectile Motion Problem
Find the x and y components of Vi.
Vi = initial velocity = 160 ft/sec
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Projectile Motion Problem
Find the x and y components of Vi.
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Projectile Motion Problem
Find the x and y components of Vi.
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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.
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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).
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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:
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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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Answers:
• Maximum height: 21.44 meters
• Total time in the air: 4.18 seconds
• Horizontal Distance: 113.7 meters