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CHAPTER 3

PROJECTILE MOTION


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Projectile motion:a body in free fall moving intwo dimensions that is subject to the forces of

gravity and air resistance and its path is aparabola

Bodies projected into the air areprojectiles.

Examples of moving body as projectile:

Shot putt High jumper Long jumper

Soccer ball Baseball Ski jumper

What are the components of the movement?


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The speed in the x-direction

is constant; in the y-

direction the object moveswith constant acceleration

g.

This photograph shows two ballsthat start to fall at the same time.

The one on the right has an initial

speed in the x-direction. It can be

seen that vertical positions of the

two balls are identical at identical

times, while the horizontal

position of the yellow ball

increases linearly.

y

x


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If an object is launchedat an initial angle of 0

with the horizontal, the analysis is similar except

that the initial velocity has a verticalcomponent.This diagram isdrawn using a

correct scale.

Which velocity ischanging as time

goes by?


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Objectives when launching projectiles:

Influence time of flight

Maximum

Minimum

Maximize horizontal displacement

Maximize vertical displacement


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Factors Influencing Projectile Trajectory

Angle of projection

Projection speed

Relative height of projection

Trajectory: the flight path of a projectile


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General shapes

Perfectly vertical (angle = ?)

Parabolic (angle = ?)

Perfectly horizontal (angle = ?)

Implications in sports

Air resistance may causeirregularities

Angle of Projection


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Range: horizontal displacement.

For oblique projection angles,

speed determines height and

range

For vertical projection angle,

speed determines height.

Projection speed:


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Difference between projection

and landing height

Greater the relative projection

height, longer the flight time,

greater the displacement.

Relative Projection Height


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Optimum Projection Conditions

Maximize the speed of projection Maximize release height

Optimum angle of projection Release height = 0, then angle = 450

Release height, then angle

Release height, then angle

Complimentary angles (A + B = 90) have the

same range Large angle higher and slower time: tennis lob,punt

Small anglelower and faster time: baseball throw


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Analyzing Projectile Motion

Initial velocity:

Horizontal component is constant

Horizontal acceleration = 0

Vertical component is constantly changing

Vertical acceleration = -9.81 m/s2


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A projectile launched with velocity will have both vertical &

horizontal components of that velocity.

Horizontal & Vertical Components:

Vertical is influenced by gravity

No force (neglecting air resistance) affects the horizontal

Horizontal relates to distance

Vertical relates to maximum height achieved

Horizontal and vertical components are independent


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Major influence of verticalcomponent

Not the horizontal component

Force of Gravity:

Constant, unchanging

Negative acceleration (-9.81

m/s2)

Apex:

Highest Point

Vertical velocity = 0 m/s

A body projected

straight upward will

have the same speedat the end of its flight

as it did when it was

launched

The Influence of Gravity


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In a vacuum, horizontal speed of a projectile

remains constant

Air resistance affects the horizontal speed of a

projectile

This class , ho rizontal veloci ty w i l l be regarded asconstant

Influence of Air Resistance


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Equations of

Constant Acceleration

Galileos Laws of constant acceleration:

v2= v1+ at

S = v1t + at2

V22= v21+ 2 aS

S = displacement; v = velocity; a = acceleration; t = time

Subscript 1 & 2 represent first or initial and second or final

point in time


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Equations of


Horizontal component : a = 0

v2= v1S = v1t

V22= v21


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Equations of


Vertical component: a = -9.81 m/s2, initial velocity of zero

(a dropped object)

v2= at

S = at2

V22= 2aS

Vertical component at apex: a = 0

0 = v21+ 2aS (det. max height)

0 = v1+ at

(total flight timemultiply by 2)


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Solving Problems Involving Projectile Motion

Projectile motion is motion with constantacceleration in two dimensions, where the

acceleration is gand is down.


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Solving Problems Involving Projectile Motion:

Procedures

1. Read the problem carefully, and choose theobject(s) you are going to analyze.

2. Drawa diagram.

3. Choosean origin and a coordinate system.

4. Decideon the timeinterval;

this is the same in both directions, and

includes only the time the object is movingwith constant acceleration g.

5. Examinethe xand ymotions separately.


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Solving Problems Involving Projectile Motion:

Procedures

6. List known and unknown quantities.

Remember that vxnever changes, and that vy=

0 at the highest point.

7. Plan how you will proceed. Use the

appropriate equations; you may have to

combine some of them.


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A movie stunt driver on a motorcycle speeds

horizontally off a 50.0-m-high cliff. How fast

must the motorcycle leave the cliff top to land

on level ground below, 90.0 m from the base of

the cliff where the cameras are? Ignore air

resistance.

Driving Off A Cliff


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A football is kicked at an angle 0 = 37.0 with a

velocity of 20.0 m/s, as shown. Calculate (a) the

maximum height, (b) the time of travel before the

football hits the ground, (c) how far away it hits the

ground, (d) the velocity vector at the maximum height,

and (e) the acceleration vector at maximum height.

Assume the ball leaves the foot at ground level, and

ignore air resistance and rotation of the ball.

A Kicked Football


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A child sits upright in a wagon

which is moving to the right at

constant speed as shown. The

child extends her hand and throwsan apple straight upward (from her

own point of view), while the

wagon continues to travel forward

at constant speed. If air resistanceis neglected, will the apple land (a)

behind the wagon, (b) in the

wagon, or (c) in front of the

wagon?

Conceptual Problem:

Where does the apple land?


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Conceptual Problem: The wrong strategy.

A boy on a small hill aims his water-balloon slingshothorizontally, straight at a second boy hanging from a

tree branch a distance daway. At the instant the water

balloon is released, the second boy lets go and falls

from the tree, hoping to avoid being hit. Show that he

made the wrong move. (He hadnt studied physicsyet.) Ignore air resistance.

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