what is kinematics? geometry of motion

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 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.

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

StartEnd

Displacement: straight line distancebetween 2 points

Start

End

The Language of Kinematics

Displacement can be measured as two components, the x and y direction:

Start

End

X displacement

Y 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 a = 10 m/s2

5 sec

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.

Horizontal Direction: • No acceleration: therefore, ax = 0.•

Vertical Direction: • Gravity affects the acceleration. It is constant and directed downward:

therefore, ay = -g.

Projectile Motion

Projectile Motion

At the maximum height:

= 0

t0

Displacement in general

Projectile Motion Formulas

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 FormulasInitial 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: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

Trigonometric Functions:

sin θ° = O / H

cos θ° = A / H

tan θ° = O / A

Trigonometric Functions:

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 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)?

Projectile Motion Problem

Find the x and y components of Vi.

Vi = initial velocity = 160 ft/sec

Projectile Motion Problem

Find the x and y components of Vi.

Projectile Motion Problem

Find the x and y components of Vi.

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