Quick Review: Four Kinematic Equations

Free Fall

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 = vi

2 + 2 a Δx

Four Kinematic Equations There are always 4 variables To use these equations you guess and check. Remember to always do 4 things:

1. Draw a diagram

2. Write what you know

3. Write what you need

4. Guess and check Let’s practice…

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.

Free Fallg is independent of 3 things:

time it’s been fallingmass of the objectif it started at rest or not

Terminal Velocity – speed when the force of air resistance is equal and opposite to the force of gravity.

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….

Part 1.Motion of Objects Projected

Horizontally

Introduction

Projectile Motion:

Motion through the air without a propulsion Examples:

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

Projectiles A projectile has only one

force acting upon - the force of gravity

Examples: golf, soccer ball, bullet, rock dropped, javelin thrower …

Factors Influencing Projectile Trajectory

Trajectory: the flight path of a projectile

Angle of projection Projection speed Relative height of

projection

Factors Influencing Projectile Trajectory

Angle of Projection General shapes

Perfectly verticalParabolicPerfectly horizontal

Implications in sports Air resistance may

cause irregularities

Factors Influencing Projectile Trajectory

Projection speed:Range:

o horizontal displacement. For oblique projection angles, speed

determines height and range.For vertical projection angle, speed

determines height.

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.

Projectile Motion

The path (trajectory) of a projectile is a parabola Describe the motion of an object in TWO

dimensionsVertical - vY

Horizontal - vX

Horizontal and vertical motion are independent (90°)

Projectile Motion

HorizontalMotion of a ball rolling freely along a level surfaceHorizontal velocity is ALWAYS constantThe horizontal component of it’s velocity does not

change. vX is constant

Projectile Motion

VerticalMotion of a freely falling objectForce due to gravityVertical component of velocity changes with time

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!

Trajectory and Range Maximum range

is at 45° Low and high

trajectory cover the same distance.

30 and 60 10 and 80 25 and…

The path (trajectory)of a projectile is a parabola

Parabolic motion of a projectile

v0

x

y

x

y

x

y

x

y

x

y

x

y

g = -9.81m/s2

• 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

ExperimentWhat do you think? Which ball will hit the ground first?

a) The left ball will hit firstb) The right ball will hit firstc) They will hit the ground at the same time.

Projectiles

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

Equations

X- Component

Y- Component

x vxi

t

y vyit 1

2at 2

vyf2 vyi

2 2ay

vyf vyi at

Note: g= 9.8 m/s^2

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?

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?

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.

Let’s try pg 99 practice D

Board Work

1. 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.

Part 2.Motion of objects projected at an

angle

vi

x

y

θ

vix

viy

Initial velocity: vi = vi [Θ]

Velocity components:

x- direction : vix = vi cos Θ

y- direction : viy = vi sin Θ

Initial position: x = 0, y = 0

x

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

a = g =

- 9.81m/s2

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?

Equations of motion:

X

Uniform motion

Y

Accelerated motionACCELERATION ax = 0 ay = g = -9.81 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

Equations

X- Component

Y- Component

x vicost

y (v i sin)t 1

2at 2

vyf2 (v i sin)2 2ay

vyf (v i sin) at

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?

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?

Projectile Motion – Final Equations

Trajectory Parabola, open down

Total time Δt =

Horizontal range Δx =

Max height hmax =

(0,0) – initial position, vi = vi [Θ]– initial velocity, g = -9.81m/s2

2 vi sin Θ

(-g)

vi 2 sin (2 Θ)

(-g)

vi2

sin2 Θ

2(-g)

PROJECTILE MOTION - SUMMARY Projectile motion is motion with a constant

horizontal velocity combined with a constant vertical acceleration

The projectile moves along a parabola

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?

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?

PROJECTILE MOTION - SUMMARYReview 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,

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