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**Quick Review: Four Kinematic Equations Free Fall**

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**Four Kinematic Equations**

Constant acceleration - an object will change its velocity by the same amount each second. You must have constant acceleration to use the four kinematic equations. Δx = ½(vi + vf) Δt vf = vi + a Δt Δx = vi Δt + ½ a(Δt)2 vf2 = vi2 + 2 a Δx

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**Four Kinematic Equations**

There are always 4 variables To use these equations you guess and check. Remember to always do 4 things: Draw a diagram Write what you know Write what you need Guess and check Let’s practice…

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Free Fall Is when an object is falling under the sole influence of gravity known as “acceleration due to gravity” = g g = 9.81m/s2 There are slight variations that are affected by altitude, we will ignore this.

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**Free Fall g is independent of 3 things: time it’s been falling**

mass of the object if it started at rest or not Terminal Velocity – speed when the force of air resistance is equal and opposite to the force of gravity.

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**Working Backwards It all works backward as well.**

If a ball is thrown straight up: It will decelerate at 9.81m/s2 At the top of it’s path the ball “hangs” in mid air. At bottom of it’s path the balls velocity is equal to vi See Diagram….

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**Part 1. Motion of Objects Projected Horizontally**

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**Introduction Projectile Motion:**

Motion through the air without a propulsion Examples:

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Projectile Motion Keep it simple by considering motion close to the surface of the earth for the time being Neglect air resistance to make it simpler

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Projectiles A projectile has only one force acting upon - the force of gravity Examples: golf, soccer ball, bullet, rock dropped, javelin thrower …

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**Factors Influencing Projectile Trajectory**

Trajectory: the flight path of a projectile Angle of projection Projection speed Relative height of projection Trajectory: the flight path of a projectile Three factors influence the trajectory of a projectile: the angle of projection, the projection speed, and the relative height of projection. Understand how these factors interact is useful within the context of a sport both for determining how to best project balls and other implements and for predicting how to best catch or strike projected balls.

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**Factors Influencing Projectile Trajectory**

Angle of Projection General shapes Perfectly vertical Parabolic Perfectly horizontal Implications in sports Air resistance may cause irregularities The angle of projection and the effects of air resistance govern the shape of a projectile’s trajectory. Angle of Projection: the direction at which a body is projected with respect to the horizontal In the absence of air resistance, the trajectory of a projectile assumes one of three general shapes, depending on the angle of projection. General shapes Perfectly vertical – projectile follows same path straight up and then straight down again Parabolic – an oblique projection angle (00-900), the trajectory is parabolic Shaped like a parabola Symmetrical, right & left halves are mirror images Perfectly horizontal – at an angle of 00, the trajectory resembles one half of a parabola. Projection angle has direct implications for success in the sport of basketball, since a steep angle of entry into the basket allows a some-what larger margin of error than a shallow angle of entry. In projection situations on a field, air resistance may, in reality, create irregularities in the shape of projectiles, trajectory.

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**Factors Influencing Projectile Trajectory**

Projection speed: Range: horizontal displacement. For oblique projection angles, speed determines height and range. For vertical projection angle, speed determines height. When projection angle and other factors are constant, the projection speed determines the length or size of a projectile’s trajectory. Projection speed: the magnitude of projection velocity For a body that is projected at an oblique angle, the speed of projection determines both the height and the horizontal length of the trajectory. The combines effects of projection speed and projection angle on the horizontal displacement, or range, of the projectile. Range: the horizontal displacement of projectile at landing The third major factor influencing projectile trajectory is the relative projection height. Relative projection height: the difference between projection height and landing height When projection velocity is constant, greater relative projection height translates to longer flight time and greater horizontal displacement of the projectile. A projectile’s flight time is increased by increasing the vertical component of projection velocity or by increasing the relative projection height.

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**Factors Influencing Projectile Trajectory**

Relative Projection Height: Difference between projection and landing height Greater the relative projection height, longer the flight time, greater the displacement. When projection angle and other factors are constant, the projection speed determines the length or size of a projectile’s trajectory. Projection speed: the magnitude of projection velocity For a body that is projected at an oblique angle, the speed of projection determines both the height and the horizontal length of the trajectory. The combines effects of projection speed and projection angle on the horizontal displacement, or range, of the projectile. Range: the horizontal displacement of projectile at landing The third major factor influencing projectile trajectory is the relative projection height. Relative projection height: the difference between projection height and landing height When projection velocity is constant, greater relative projection height translates to longer flight time and greater horizontal displacement of the projectile. A projectile’s flight time is increased by increasing the vertical component of projection velocity or by increasing the relative projection height.

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**Projectile Motion The path (trajectory) of a projectile is a parabola**

Describe the motion of an object in TWO dimensions Vertical - vY Horizontal - vX Horizontal and vertical motion are independent (90°)

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**Projectile Motion Horizontal**

Motion of a ball rolling freely along a level surface Horizontal velocity is ALWAYS constant The horizontal component of it’s velocity does not change. vX is constant

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**Projectile Motion Vertical Motion of a freely falling object**

Force due to gravity Vertical component of velocity changes with time

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Package drop The package follows a parabolic path and remains directly below the plane at all times The vertical velocity changes (faster, faster) The horizontal velocity is constant!

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**Trajectory and Range Maximum range is at 45°**

Low and high trajectory cover the same distance. 30 and 60 10 and 80 25 and…

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**The path (trajectory) of a projectile is a parabola**

Parabolic motion of a projectile

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y v0 x

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y x

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y x

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y x

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y x

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**g = -9.81m/s2 y y-motion is accelerated**

Acceleration is constant, and downward a = g = -9.81m/s2 The horizontal (x) component of velocity is constant The horizontal and vertical motions are independent of each other, but they have a common time g = -9.81m/s2 x

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**Experiment What do you think? Which ball will hit the ground first?**

a) The left ball will hit first b) The right ball will hit first c) They will hit the ground at the same time.

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Projectiles

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**Both balls hit the ground at the same time. Why?**

As soon as both balls are released by the launcher, they are in "freefall. The only force acting on both objects is gravity. Both objects accelerate at the same rate, 9.8m/s2 Both objects covering the same distance at the same rate and therefore hit the ground at the same time

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Equations X- Component Y- Component Note: g= 9.8 m/s^2

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**ANALYSIS OF MOTION ASSUMPTIONS:**

x-direction (horizontal): uniform motion y-direction (vertical): accelerated motion no air resistance QUESTIONS: What is the trajectory? What is the total time of the motion? What is the horizontal range? What is the final velocity? What is the initial velocity?

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**Example: Projectiles launched horizontally**

What is the total time of the motion? What is the horizontal range? What is the final velocity? What is the initial velocity? The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas River. Suppose you kick a rock horiaontally off the bridge. The magnitude of the rock’s horizontal displacement is 45m How long does it take the rock to hit the ground? What speed did you have to initially have to kick the rock? How fast was the rock going before hitting the ground?

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**Example: Projectiles launched horizontally**

What is the total time of the motion? What is the horizontal range? What is the final velocity? What is the initial velocity? People in movies often jump from buildings into pools. If a person jumps horizontally from the 10th floor(30m) to a pool that is 5m away from the building, how long does it take for him to hit the water in the pool? What initial speed must the person jump to make it? What is the final velocity of the person before he hits the water’s surface.

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Let’s try pg 99 practice D

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Board Work Erica kicks a soccer ball 12 m/s at horizontally from the edge of the roof of a building which is 30.0 m high. 2. A ball thrown horizontally from the roof of a building lands 36m from the base of the building. Just before impact the ball had a velocity of 25m/s. 3. A boy kicked a can horizontally from a 6.5 m high rock with a speed of 4.0 m/s. 4.A car drives straight off the edge of a cliff that is 54 m high. The police at the scene of the accident note that the point of impact is 130 m from the base of the cliff.

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**Part 2. Motion of objects projected at an angle**

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x y vi Initial position: x = 0, y = 0 vix viy Initial velocity: vi = vi [Θ] Velocity components: x- direction : vix = vi cos Θ y- direction : viy = vi sin Θ θ

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**y a = g = - 9.81m/s2 x Motion is accelerated**

Acceleration is constant, and downward a = g = -9.81m/s2 The horizontal (x) component of velocity is constant The horizontal and vertical motions are independent of each other, but they have a common time x

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**ANALYSIS OF MOTION: ASSUMPTIONS**

x-direction (horizontal): uniform motion y-direction (vertical): accelerated motion no air resistance QUESTIONS What is the trajectory? What is the total time of the motion? What is the horizontal range? What is the maximum height? What is the final velocity?

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**X Uniform motion Y Accelerated motion**

Equations of motion: X Uniform motion Y Accelerated motion ACCELERATION ax = 0 ay = g = m/s2 VELOCITY vx = vi cos Θ vy = vi sin Θ + a t DISPLACEMENT Δx = vi cos Θ t Δy = vi sin Θ t + ½ a t2

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Equations X- Component Y- Component

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**Example: Projectiles launched @ an angle **

Erica kicks a soccer ball 12 m/s at an angle of 40 degrees above the horizontal. *Don’t forget to draw your chart* What are the x and y components of the vi? How long does it take the ball to hit the ground? What is the max height the ball travels? How far does she kick the ball?

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**Example: Projectiles launched @ an angle **

An archer needs to be sure to shoot over the wall of the castle. He raises his bow at an angle of 65° and fires his arrow with an initial velocity of 43m/s. *Don’t forget to draw your chart* What are the x and y components of the vi? How long does it take the arrow to hit the ground? What is the max height the arrow travels? How far does the archer shoot the arrow?

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**Projectile Motion – Final Equations**

(0,0) – initial position, vi = vi [Θ]– initial velocity, g = -9.81m/s2 Trajectory Parabola, open down Total time Δt = Horizontal range Δx = Max height hmax = 2 vi sin Θ (-g) vi 2 sin (2 Θ) (-g) vi2 sin2 Θ 2(-g)

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**PROJECTILE MOTION - SUMMARY**

Projectile motion is motion with a constant horizontal velocity combined with a constant vertical acceleration The projectile moves along a parabola

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**The monkey and the zookeeper!!**

A golfer practices driving balls off a cliff and into the water below. The dege of the cliff is 15m above the water. If the golf ball is launched at 51m/s at and angle of 15°, how far does the ball travel horizontally before hitting the water?

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**The monkey and the zookeeper!!**

A zookeeper finds an escaped monkey hanging from a light pole. Aiming her tranquilizer gun at the monkey, she kneels 10m away from the light pole, which is 5m high. The tip of her gun is 1m above the ground. At the same moment that monkey drops a banana, the zookeeper shoots. If the dart travels at 50m/s, will the dart hit the monkey, the banana, or neither one?

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**PROJECTILE MOTION - SUMMARY**

Review for Test 2 Pg 109 # 2, 3, 6, 12, 13, 14, 15, 17, 18, 20, 21, 24, 25, 27, 28, 30, 31, 32, 34, 37 Pg 69 # 18, 20, 22, 24, 26, 30, 31, 33, 35, 38, 39, 46

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Motion in Two Dimensions

Motion in Two Dimensions

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