05 - projectile motion.pptx

7/29/2019 05 - Projectile Motion.pptx

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Announcements

Consultation Hours:

TWTh 11:30-2:45


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Projectile Motion

Ms. Mikaela Irene Fudolig

Physics 71


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What is a projectile?

A projectile is any body that is given an

initial velocity (may be zero) and then

follows a path determined entirely by the

effects ofgravitational acceleration and air

resistance.

This path is called the projectiles trajectory.

For now, we will neglect air resistance.


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Projectile Motion in 2D

Suppose that at any time t, we can find the

particle at the point (x(t),y(t))

+y

+x

(x(t),y(t))


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Then, its position vector is given by:

+y

+x

(x(t),y(t))

( ) ( ) ( )r t x t i y t j


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Since the i andj components are

independent, its velocity vector is given by:

+y

+x

(x(t),y(t))

( ) ( ) ( )x yv t v t i v t j


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So, its acceleration vector is given by:

+y

+x

(x(t),y(t))

( ) ( ) ( )x y

a t a t i a t j


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The i andj components, being

independent, are obtained separately from

two different sets of kinematic equations:

0

2

0 0

2 2

0

0

1

22

2

y y y

y y

y y y

y y

v v a t

y y v t a t

v v a y

v vy

t

0

2

0 0

2 2

0

0

1

2

2

2

x x x

x x

x x x

x x

v v a t

x x v t a t

v v a x

v vx

t


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2D motion can be

treated as motion in thex-direction and motion in

the y- direction occurringSIMULTANEOUSLY!


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Recall: a projectile is only affected by

gravity (and air resistance, which we

neglect).

What is the projectiles acceleration?

2 9.81 ma j gjs


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Therefore:

0x

y

a

a g


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Idea of Projectile Motion

Movement in the

x- and y-directions

are TOTALLY

INDEPENDENT of

each other!

a=-9.81m/s2

unequallyspaced


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Movement in the

x- and y-directions

are TOTALLY

INDEPENDENT of

each other!equallyspaced

notaccelerating(a=0)


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Movement in the

x- and y-directions

are TOTALLY

INDEPENDENT of

each other! equallyspaced

unequallyspaced


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Movement in the

x- and y-directions

are TOTALLY

INDEPENDENT of

each other!


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Motion ofBLUE ball

+ Motion ofYELLOW ball

Motion ofRED ball

y-motion

x-motion

gravityv0y

v0x


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Example 1a: Comparison of Final

Velocities

Ball A, initially 1m above

the ground, is thrown

downwards with an initial

velocity

and hits the ground tA

seconds later, with a

velocity

Ball B, initially 1m above



velocity

and hits the ground tB


velocity

0 0

A yv v j

A Ax Ayv v i v j

0 0 0 B x yv v i v j

B Bx Byv v i v j

Compare vAy

and vBy

.


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Velocities

v0y

t=tA

t=tB

vA vB

y=1m

y=0

v0y

v0x

vB0

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j


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Velocities

t=tA

t=tB

y=1m

y=0

v0y

v0y

v0x

vB0

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j

vA vB


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Velocities

t=tA

t=tB

y=1m

y=0

v0y

v0y

v0x

vB0

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j

vA vB

|vAy|=|vBy|


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Example 1b: Comparison of Final

Velocities




velocity



velocity




velocity



velocity

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j

Compare vAx

and vBx

.


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Velocities

t=tA

t=tB

y=1m

y=0

v0y

v0y

v0x

vB0

vx

vA vB

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j


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Velocities

t=tA

t=tB

y=1m

y=0

v0y

v0y

v0x

vB0

vx

vA vB

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j


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Velocities

t=tA

t=tB

y=1m

y=0

v0y

v0y

v0x

vB0

vx=v0x

vA vB

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j


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Velocities

t=tA

t=tB

y=1m

y=0

v0y

v0y

v0x

vB0

vx=v0x

vA vB

vAx=0

vBx= v0x>vAx


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Conceptual Exercise 1




velocity



velocity




velocity



velocity

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j

Which ball hits the ground earlier?


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The Basic Idea

Treat motion in the x-

direction and motion inthe y- direction as

INDEPENDENT and

SIMULTANEOUS!


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velocity



velocity



upwards with an initial

velocity



velocity

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j

Which ball hits the ground earlier?


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velocity



velocity



upwards with an initial

velocity



velocity

0 0

A yv v j

A Ax Ayv v i v j


B Bx Byv v i v j

Compare vAy and vBy.


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An object is thrown with an initial speed

1m/s at an angle 30 above the horizontal.

What is the objects speed at the top of its

flight?


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The Basic Idea

Treat motion in the x-

direction and motion inthe y- direction as

INDEPENDENT and

SIMULTANEOUS!


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Some extra terms

The range is the total horizontal

displacement of the projectile.

The maximum height is the maximum

elevation that the object attains while in

flight.


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Homework (Reading Assignment)

Show that if the INITIAL and FINAL

elevations are the SAME, then the

maximum range occurs when the

projectile is thrown at an angle of 45degrees above the horizontal.


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Prove:

That the trajectory of a projectile moving

under the sole influence of gravity is a

PARABOLA.

HINT: Show that the y(x) is a parabola.


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Numerical Example 1

An object is thrown with a speed 1m/s at

an angle 30 above the horizontal.

What is the horizontal component of the initial

velocity?

What is the vertical component of the initial

velocity?


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Numerical Example 1

An object is thrown with a speed 1m/s at

an angle 30 above the horizontal. If the

initial and final elevations are the same:

What is the total time of flight?

What is the range of the projectile?

What is the maximum height reached by the

projectile?Draw the x vs. t, vx vs. t, ax vs. t, y vs. t, vy vs. t,

and ay vs. t diagrams.


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Numerical example 2

A daring 510-N swimmer dives off a cliff

with a running horizontal leap. What must

her minimum speed be just as she leaves

the top of the cliff so that she will miss theledge at the bottom, which is 1.75m wide

and 9.00m below the top of the cliff?


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Numerical Example 3

A ball is thrown at an angle 15 at some

height h above the ground. It is observed

that the ball landed 30 m away from the

building. If it took the ball 5.0s to land onthe ground, what is the height h?

05 - projectile motion.pptx

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