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Special Cases of Force

Projectile Motion

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Projectile Motion

• A projectile is any object which once a force is used to throw it, hit it , propel it in some fashion, no other force acts on the object except for gravity.

• Projection can be – horizontal, with no initial vertical velocity– vertical, with some initial vertical velocity

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Projectile Motion: A special case of uniformly accelerated motion

If air resistance is negligible then only gravity affects the path (trajectory) of a projectile.

This path is a parabola.

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Characteristics of Projectile Motion

• The motion’s dimensions are vectors.• The projectile will move along it’s trajectory

(path) in both the horizontal (x) direction and the vertical (y) direction at the same time.

• The two directions are independent of each other.• Only time is the same between the horizontal and

vertical motion. It is, after all, only one object.

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Independent Motions

Horizontal

Vertical

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Real Motion is the Combination of the Horizontal and Vertical Motions

The blue dot show the real motion

The path taken is the trajectory

This is horizontal projection

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Initial Velocity – V Along the Trajectory

• Angular projection - The initial velocity is the resultant of adding the two vector quantities together.

• The projection includes an angle of projection

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Initial Velocity has an X and Y Component

• The vertical component and the horizontal component are independent of each other.

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Horizontal and Vertical Components of VelocityVertical velocity decreases at a constant ratedue to the influence of gravity. It becomes zero. Then increases in the negative direction

Positive velocitygets smaller

Vertical velocity = 0

Negative velocitygets larger

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Calculating ComponentsYou have learned to calculate components of a vector when

we looked at inclined planes.

The components are calculated by using the trig functions.

The initial velocity acts as the hypotenuse of the right triangle.

Vi

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Continued Calculation

• The vertical velocity and the horizontal velocity are the legs of the triangle.

Calculate the Vy using sin = o/h

Calculate Vx using cos = a/h

Vy

Vx

Vi

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Horizontal Displacement of Projectiles• Horizontal projection or projection at an angle

– graph of horizontal displacement v time

As time progresses, the object gets further and further from its starting point.

Time

Displacement

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Vertical Displacement of Projectiles

• Vertical Projection and Projection at an Angle – graph of vertical displacement (height) v time

The object rises into the air and then return to earth.

Time

Height

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Vertical Displacement of Projectiles

• Horizontal projection– Graph of vertical displacement (height) vs time

Time

Height

The object leaves a horizontal surface and fall to the ground.

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Horizontal Velocity of Projectiles

• Horizontal Projection and Projection at an Angle

The horizontal velocity of a projectile remains constant from the time it is projected until gravity brings it to the ground.Remember: We are using an ideal situation where there is no air resistance

Time

Velocity

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Vertical Velocity of Projectiles

• Vertical Projection and Projection at an Angle

Time

Velocity

0

For objects projected directly upward or projected at some angle above the ground, the vertical velocity must begin positive, decrease to zero, and then increase in the negative direction. (Remember, gravity is negative)

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Vertical Velocity of Projectiles

• Horizontal projection

Time

Velocity

0

Horizontal projection begins with an initial vertical velocity of zero.The vertical velocity then increases in a negative direction.

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Horizontal Acceleration of Projectiles• Since we are idealizing the projection, we

do not take into account any air resistance.• We can, therefore, say there is no horizontal

acceleration.

Time

Acceleration

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Vertical Acceleration of Projectiles

• Vertical acceleration is the result of the pull of gravity, (-9.8 m/s2)

• This is the same on the way up and on the way down.

Time

Acceleration

0

-9.8

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Important Notes on Vertical and Angular Projection

• The range (x) is the farthest the object will travel horizontally.

• The maximum height (ymax ) is the farthest the object will travel vertically.

• Y equals zero when it is at its lowest point.

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Determining the Range

You can determine the displacement (range) of a projectile any any point along the trajectory.

tvx xX = horizontal distance (range)

vx = horizontal velocity

t = time

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Determining the Height

• You can determine the height (y) at any point in the trajectory!

2

2

1gttvy y

y = vertical displacement

vy = initial vertical velocity

g = acceleration due to gravity

t = time

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Other Important Notes on Vertical and Angular Projection

• At the highest point of the trajectory, it is the exact midpoint of the time.

• It takes the projectile half of the time to get to the top.

• When the projectile gets to the top, it has to stop going up and start going down, so the velocity in the y-direction at the highest point is zero for a split second.

• As the projectile falls, it is in free fall.

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• Projection angle

– aka release angle or take-off angle• Projection velocity

– aka initial or take-off velocity

• Projection height – aka above or below landing

3 Primary Factors Affecting Trajectory

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Projection Angle

• The optimal angle of

projection is dependent on

the goal of the activity.

• For maximum height, the optimal angle is 90o.

• For maximum distance, the optimal angle is 45o.

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The effect of Projection angle on the Range of a projectile

The angle that maximizes Range is = 45 degrees

10 degrees30 degrees40 degrees45 degrees60 degrees75 degrees

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The effect of Projection velocity on the Range of a projectile

• 10 m/s @ 45 degrees Range ~ 10 m• 20 m/s @ 45 degrees Range ~ 40 m• 30 m/s @ 45 degrees Range ~ 90 m

1009080706050403020100

0

10

20

30

40

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Projectile Problems

• Ignore air resistance.

• ay = g = -9.81 m/s2

• Set up the two dimension separately

Origin x Origin y

Positive x Positive y

xi = constant yi =

vxi = vyi =

ax = 0 ay = g

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Projectile Problems – Two Dimensional Kinematics

• Write general kinematic equations for each direction

• Rewrite them for the problem at hand• Find the condition that couples the horizontal and

vertical motions (usually time)

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Equations of Constant Acceleration

• x = vxt

• d = ½ (vyf + vyi)t

• d = vyit + ½ gt2

• vvf2 = vvi

2 + 2gd

These 3 equations are for the vertical part of the motion

This is the only equation to use for the horizontal part of the motion

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The Monkey and the Banana

A zookeeper must throw a banana to a monkey hanging from the limb of a tree. The monkey has a habit of dropping from the tree the moment that the banana is thrown. If the monkey lets go of the tree the moment that the banana is thrown,will the banana hit the monkey?

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When you take gravity into consideration you STILL aim at the monkey!

It works! Since both banana and monkey experience the same acceleration each will fall equal amounts below their gravity-free path. Thus, the banana hits the monkey.

Monkey’sGravity free path is “floating” at height of limb

Banana’s Gravity free path

Fall thru same height

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Homework

• Chapter 6 read to page 152

• Problems: page 164, # 33,35,51,52,53,56

• 57,60.