Kinematics

Unit 8

POE Ballistic Device

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.

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: Distance = 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. d = 1 mile

The Language of Kinematics

Displacement (s): Vector Quantity

• A measure of an object’s position measured from it’s original position or a reference point.

• The terms displacement and distance are used interchangeably, although not always correctly

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 displacement divided by the time.

• The terms velocity and speed are sometimes used interchangeably, although not always correctly.

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

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 divided by the elapsed time (change in time)

The Language of Kinematics

Acceleration (a): Vector Quantity

• Ex: Assume that a car, who starts at rest, is going 50 m/s (meters per second) after 5 seconds. What is it’s average acceleration?

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 Assumptions

To start:

• 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 Assumptions

Projectile Motion Assumptions

At the maximum height:

= 0

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 to:

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!

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, it lands

after 8 seconds.

Projectile Motion Problem

• Find the x and y components of Vi.

•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

• What is the ball’s range (the distance traveled horizontally)?

Projectile Motion Problem-2You try one:

A golf ball is hit at an angle of 20 degrees from the ground, with an

initial velocity of 100ft/sec. It lands on the ground after 3 seconds.

Answers:

• Horizontal Distance: 281.91 ft

Projectile MotionThe Ballistic Device

Projectile MotionThe Ballistic Device

Objective:

Create a device that will toss a projectile (ping-pong ball)

accurately within a given range

Constraints:

• Range between 5 and 15 feet

• Fit inside a 1 ft x 1 ft footing

• No high-power pressure gasses or combustibles

• Constructed from found materials

Projectile MotionThe Ballistic Device

Final Test:

Land in a target specified by the teacher on day of test

Projectile MotionThe Ballistic Device

Method:

• Calculate initial Velocity (Vi), and assume it stays constant throughout the test

• Calculate resulting range for specified angles

• Plot range vs angle and use to predict angle for specified range

Projectile MotionThe Ballistic Device

Calculate Initial Velocity:

• Pick an angle

• Shoot projectile 10 times at chosen angle and calculate the mean range

• Use the angle, mean range and gravity constant to calculate initial velocity

Projectile MotionThe Ballistic Device

Projectile MotionThe Ballistic Device

Finding Formula for Initial Velocity :

Remember?

Both involve time (t) which is extremely difficult to measure accurately

and

Projectile MotionThe Ballistic Device

For entire motion, total vertical displacement = 0, therefore y = 0.

Finding Formula for Initial Velocity :

Projectile MotionThe Ballistic Device

Finding Formula for Initial Velocity:

Projectile MotionThe Ballistic Device

Finding Formula for Initial Velocity:

Projectile MotionThe Ballistic Device

Finding Formula for Initial Velocity:

Projectile MotionThe Ballistic Device

Trigonometric Identity:

Finding Formula for Initial Velocity:

Projectile MotionThe Ballistic Device

Finding Formula for Initial Velocity:

Projectile MotionThe Ballistic Device

Finding Formula for Range knowing Initial Velocity and Angle:

Projectile MotionThe Ballistic Device

Finding Formula for Range knowing Initial Velocity and Angle:

Projectile MotionThe Ballistic Device

Finding Formula for Range knowing Initial Velocity and Angle:

Using this formula, the range (x) can be calculated for various angles