2-Dimensional Motion - Projectiles - PowerPoint Presentation

Slide1

2-Dimensional Motion - Projectiles

Now it starts to get more

interesting

(and don’t get freaked out by the equations and subscripts)Slide2

Projectiles – What path do they follow?

http://www.us-inauguration-day-2009.com/human_cannonball.jpgSlide3

Projectiles follow parabolic paths

Most important thing

to remember is that

horizontal and vertical

motion are independent

of one another.

From now on,

Horizontal = X direction

Vertical = Y directionSlide4

Let’s look at the horizontal and vertical components individually

Which way does gravity point? DOWN!!!

So, there is no gravity in the horizontal direction (x-direction)

There is only gravity in the vertical direction (y-direction)

So, in general, there is no acceleration in the horizontal direction (x-direction)

Take a moment to let that sink in.

This is where parabolic motion comes from. Why? Let’s find out…Slide5

What is the X-component of motion?

Same as ‘missing acceleration’ case for one-dimensional motion.

X = V

0

T

But since we have 2 dimensions, we want to distinguish further between X and Y, so

X = V

0xT“V0

” = “V naught” = same thing as “V initial”

This is how the book writes it, so I don’t want you to get confusedSlide6

Now let’s look at the Y-direction

Y direction has gravity

So, with no initial vertical speed, the position in the y-direction follows the free fall equation:

Y = ½ gt

2

However, there will be cases where we have an initial vertical speed

Y =

V0yt+ ½ ay

t

2

=

V

0y

t +

½ gt

2

, where g = 9.8m/s

2Slide7

So, let’s bring it together

X stuff Y stuff_______________

X =

horiz

position Y =

vert

position

Ax = accel in x-dir Ay =

accel

in y-dir

V

x

= velocity in x-dir

V

y

= velocity in y-dir

V

0x

= Init

veloc

in x-dir V

0y

= Init

veloc

in y-dir

V

fx

= final

veloc

in x-dir

V

fy

= final

veloc

in y-dir

T = time T = timeSlide8

All the 1-D equations you know and love work in 2–D!

Just use subscripts!

When once we had… …Now we have

v =

a∙t

v

x = axt, v

x

= v

0x

+

a

x

t

x = ½ at

2

x = ½ a

x

t

2

, x = v

0x

t+

½ a

x

t

2

v

f

2

= v

i

2

+ 2ax v

fx

2

= v

ix

2

+ 2a

x

xSlide9

And the same for the Y-direction

Just use subscripts!

When once we had… …Now we have

v =

a∙t

v

y = ayt, v

y

= v

0y

+

a

y

t

y = ½ at

2

y

= ½ a

y

t

2

,

y

= v

0y

t+

½ a

y

t

2

v

f

2

= v

i

2

+ 2ay v

fy

2

= v

iy

2

+ 2a

y

x

And remember that nine times out of ten, the acceleration in the y-direction (a

y

) = g = 9.8m/s

2Slide10

So then why is projectile motion parabolic?

Because of the interaction between X and Y components of motion

Even though they are independent, the way in which they work together yields parabolic motion

When there is acceleration in the y-direction (gravity) and NO acceleration in the x-direction, you have equation of the form x = f(t) and y = f(t

2

)

x = v

0x t and y = v0y

t+ ½ a

y

t

2

Slide11

Now, Let’s look at some projectiles

http://media.photobucket.com/image/parabolic%20motion/Finer_Kitchens/Marilyn_CakeBalls/scan0008.jpgSlide12

Let’s look at the velocity vectors – what do you notice?

http://www.phys.ttu.edu/~rirlc/Lecture6.htmlSlide13

Examine the two different components of the velocity – X vs. Y

http://www.phys.ttu.edu/~rirlc/Lecture6.html

First, note the launch

angle

θ

0

The initial horizontal (X) component of V is given by

Vcos

(

θ

)

The initial

vertical (Y)

component of V is given by

Vsin

(

θ

)Slide14

Examine the two different components of the velocity – X vs. Y

http://www.phys.ttu.edu/~rirlc/Lecture6.html

Now note that the vertical (Y) component of motion changes

Horizontal (X) component stays the same

Because Y component changes, Velocity vector changes both direction and magnitude during flightSlide15

Now let’s look at some animations

For motorcycle and archery fun, let’s go to…

http

://www.mhhe.com/physsci/physical/giambattista/proj/projectile.html