Chapter 3

Two-Dimensional Motion Projectiles launched at an angle

Some Variations of Projectile Motion

   An object may be fired horizontally The initial velocity is all in the x-direction  v i = v x and v y = 0 All the general rules of projectile motion apply

Projectile Motion at an angle

How are they different?



Projectiles Launched Horizontally

– – The initial vertical velocity is 0.

The initial horizontal velocity is the total initial velocity.



Projectiles Launched At An Angle

– – – Resolve the initial velocity into

x

and

y

components.

The initial vertical velocity is the

y

component.

The initial horizontal velocity is the

x

component.

Some Details About the Rules

 x-direction • • • • a x = 0 v x = v ix = v i cos Θ i = constant x = v i x t • This is the only equation in the x-direction since there is constant velocity in that direction Initial velocity still equals final velocity

More Details About the Rules

 y-direction  v iy = v i sin Θ i  Free fall problem  a = g  Object slows as it goes up (-9.8m/s 2 )    Uniformly accelerated motion, so the motion equations all hold JUST LIKE STOMP ROCKETS Symmetrical

Problem-Solving Strategy



Resolve

the initial velocity into x- and y components 

Treat

the horizontal and vertical motions independently  Make a

chart

vertical motion again, showing horizontal and 

Choose

to investigate up or down.



Follow

rules of kinematics equations

Solving Launched Projectile Motion v i = (r , Θ) = (v ix , v iy )

a = v i = v f = t = x = Horizontal 0

v ix v ix

# # = range a = v i = v f = t = x = Vertical +/- 9.8m/s 2 = g

0 or v iy v iy or 0

# Max height = y

UP or DOWN INVESTIGATION… Where do the resolved components go?

Projectile Motion at an angle

Example 1: The punter for the Steelers punts the football with a velocity of 27 m/s at an angle of 30  . Find the ball’s hang time, maximum height, and distance traveled (range) when it hits the ground. (Assume the ball is kicked from ground level.) Looking for: Total time (t) Max height (y) Range (x) Given: v i = (27m/s, 30 o )

What do we do with the given info?

v i = (27m/s, 30 o ) v i = ( 23.4m/s , 13.5m/s ) “resolved” vector 27m/s What are the units?

m/s

30 o

V ix = 27cos30 V ix = 23.4

V ix = 23.4m/s V iy = 27sin30 V iy = 13.5

V iy = 13.5m/s

So where does this info “fit” in the chart?

a = v i = v f = t = x = Horizontal 0

23.4m/s 23.4m/s

Vertical a = v i =

V iy if solving “up” = 13.5m/s

v f =

V iy if solving “down” = 13.5m/s

t = x =

Pick a “side” to solve – symmetry

Up:

a = v i = v f = t = x = Horizontal 0

23.4m/s 23.4m/s v f 2 = v i 2 + 2gy y = 9.3m

Vertical a =

-

9.8m/s 2 v i =

13.5m/s

v f =

0

t = y =

v f = v i + gt t = 1.38s

Projectile Motion at an angle

Pick a “side” to solve – symmetry

Down:

a = v i = v f = t = x = Horizontal 0

23.4m/s 23.4m/s v f 2 = v i 2 + 2gy y = 9.3m

Vertical a = 9.8m/s 2 v i =

0

v f =

13.5m/s

t = y =

v f = v i + gt t = 1.38s

Projectile Motion at an angle

On the horizontal side

 Max height occurs midway through the flight.

 We found t = 1.38s both directions (up and down).

 How long is the projectile in the air?

DOUBLE this time for total air time t = 1.38 x 2 = 2.76s

 What about range?

x = vt x = (23.4)(2.76) x = 64.6m =

RANGE

This tells us…

 Now we only need an initial velocity vector to determine all of the information we need to have a detailed description of where an object is in its path.

v i = (r , Θ) = (v ix , v iy )

Maximum Range vs. Maximum Height

 What angle of a launched projectile gets the maximum height ?

90 o

 What angle of a launched projectile gets the maximum range ?

45 o

Projectile Motion at Various Initial Angles

  Complementary values of the initial angle result in the same range – The heights will be different The maximum range occurs at a projection angle of 45 o

Non-Symmetrical Projectile Motion

  Follow the general rules for projectile motion Break the y-direction into parts – up and down – symmetrical back to initial height and then the rest of the height

your homework …

 We are going to see what kind of job Hollywood writers and producers would do on their NECAP assessments…  Watching a clip of the Bus Jump, use the timer provided to time the flight of the bus and then do the actual calculations for homework.