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Chapter 7 Projectile motion

Two

Kinematics in Dimensions

Projectile Motion

- Projectile motion is motion in two directions
- Motion in the x-direction is independent of the

y-direction - Motion in the y-direction is independent of the

x-direction

What is a Projectile?

- A projectile is any object that is placed into

free flight and is being affected by gravity. - The path of a projectile is called the

trajectory.

Trajectory

More on Trajectories

- The trajectory of a projectile forms the shape of

a parabola.

Parabola

Different Trajectories

- Depending on initial location and angle,

projectiles can form different paths, all of

which possess some parabolic shape.

This path is called a half parabola.

The Half Parabola

- An object is launched horizontally and allowed to

fall. - There is no vertical velocity at the start, v1y

is zero. - The time of flight is equal to the time it would

take to drop the object from rest.

Different Trajectories

- Depending on initial location and angle,

projectiles can form different paths, all of

which still possess some parabolic shape.

This path is called a partial parabola.

Different Trajectories

- Depending on initial location and angle,

projectiles can form different paths, all of

which still possess some parabolic shape.

This path is called a full parabola.

Full Parabola

The Full Parabola

- The key to the full parabola is symmetry.
- Try to identify some points of symmetry.

Throw

Quick Comparison of Paths

- Half Parabola
- Launched Horizontally from some height
- Full Parabola
- Launched at an angle from ground level
- Symmetrical trajectory
- Partial Parabola
- Launched at an angle from some height

Half Parabola Timing

- The time of flight of a half parabolic path is

equal to that of simply dropping the object from

the same height.

Horizontal velocity (vx) has no affect on flight

time because it is not affected by gravity.

Jill drops the yellow ball off of a cliff. What

happens to the ball? Does it have constant

velocity?

Now Jill drops the yellow ball and throws the red

ball horizontally. Which ball will hit the

ground first?

Half Parabola Summary

- Objects must be dropped from some height d1y.
- The vertical reference point is usually the

ground or floor. - The time of projectiles flight is identical for

that of simply dropping the object the same

distance (straight down). - Horizontal velocity remains constant in all

projectile problems (v1x v2x).

- A stone is thrown horizontally from the top of a

78.4m high cliff at 5m/s. - a) How long does it take to reach the bottom?

Var X Y Want

a

v1

v2

d1

d2

t

- A stone is thrown horizontally from the top of a

78.4m high cliff at 5m/s. - a) How long does it take to reach the bottom?

- A stone is thrown horizontally from the top of a

78.4m high cliff at 5m/s. - b) How far from the base does it land?

- A stone is thrown horizontally from the top of a

78.4m high cliff at 5m/s. - c) What are the final vy and vx ?

Sample Partial

- A cannon nestled in the side of a cliff (d1y

65m) fires a cannon ball at 26 . How long

until the ball splashes into the sea?

Fire

Sample Problem

- A toy car is raced off a table (1.1m high) onto

the floor below. - How long did it take for the car to crash on the

floor?

Object launched horizontally Multi media studio

What do you notice about the horizontal velocity

in each of the following animations?

- Horizontally launched projectiles
- The plane and the package
- The truck and the ball

Projectile Motion Type

Not all object are launch horizontally

Objects can be launched at an angle

- Recall the trajectory of the golf ball when hit

with a 3 iron. - What would the trajectory of a 9 iron look like?
- The loft of the club changed the launch angle.

- Object 1 was launched at 60o
- Object 2 was launched at 30o

- Object 1 was launched from a 25m high cliff at 0o
- Object 2 was launched at 60o

Projectile motion generator

Analytical Vector Addition

- You will need to understand the basic trig

functions.

SOH

CAH

hyp

opp

TOA

adj

Initial Velocity Breakdown

- When an object is launched at some angle, its

initial velocity (v1) can be broken down into two

components. - Horizontal Component (Vx)
- Vertical Component (Vy)
- What shape is formed?
- Consider also the launch angle (q).

Please Note horizontal and vertical components

are independent of one another. The only

commonality is time.

Right Triangle

q

Initial Velocity Breakdown (Cont.)

- Consider the breakdown from the previous slide

again. - There are trigonometric relationships between the

sides and angles of a right triangle.

q

Practicing Trig Functions

- Consider the triangle below.
- Solve for the unknown values.

Searching for x

Searching for y

Sample Velocity Breakdown

- A dart gun is fired at an angle of 30 with a

muzzle velocity of 40m/s. - Calculate the components of the velocity?

Horizontal Component (x)

Vertical Component (y)

q

Make sure your calculator is in Degree mode!

Object launched at an angle Multi media studio

- Parabolic motion of projectiles
- Non-Horizontally launched projectiles
- Maximum Range
- Monkey and the Zoo Keeper

- Given an initial velocity of 40m/s and an angle

of 25 - find v1x v1y

Searching for y

Searching for x

Sample Full Parabola Problem

- A golf ball is struck at an angle of q 36 with

the horizontal at a velocity of 45m/s. - What are the components of the velocity (v1x and

v1y)?

Horizontal Component

Strike

Vertical Component

q

Practice Problems

- Homework
- WS 7b 1-2
- WS 7c 1-3

X Y are Independent

Problem Solving Strategies

- Solve for the horizontal component Vxi
- Use trig functions
- Solve for the vertical component Vyi
- Use trig functions
- Solve each direction (x y) separately
- Symmetry can be used when the launching landing

places are the same height.

A football player kicks a ball at 27m/s at an

angle of 30.

- Find the hang time
- find the horizontal distance the ball travels.
- The maximum height of the ball.

Horizontal Horizontal Horizontal Vertical Vertical Vertical

V K W V K W

a g -9.8

V1x v1y

v2x v2y

dx dx dy dy

t t t

Problem Solving Strategies

- Step 1 Solve for the horizontal and vertical

components (V1x V1y )

V27m/s

V1y ?m/s

V1x ?m/s

Searching for y

Searching for x

Problem Solving Strategies

- Symmetry can be used when the launching landing

places are the same height.

Vy

15.0m/s

12.5m/s

10.0m/s

7.50m/s

5.00m/s

2.50m/s

0.00m/s

A football player kicks a ball at 27m/s at an

angle of 30.

- Find the hang time
- find the horizontal distance the ball travels.
- The maximum height of the ball.

Horizontal Horizontal Horizontal Vertical Vertical Vertical

V K W V K W

a g -9.8

v1x 23.4 V1y 13.5

v2x 23.4 v2y

dx dx dy dy

t t t

A football player kicks a ball at 27m/s at an

angle of 30.

- Find the hang time

Symmetry

A football player kicks a ball at 27m/s at an

angle of 30.

b) Find the horizontal distance

A football player kicks a ball at 27m/s at an

angle of 30.

c) Find the maximum height

What is true about the vertical velocity at the

maximum height?

Vy

Vy0m/s

15.0m/s

12.5m/s

10.0m/s

7.50m/s

5.00m/s

2.50m/s

0.00m/s

- A football player kicks a ball at 27m/s at an

angle of 30. Find the max height

An arrow is shot at 44m/s at an angle of 60

- Find the maximum height of the arrow.
- Find the horizontal distance the arrows travels.
- Find the hang time

Problem Solving Strategies

- Step 1 Solve for the horizontal and vertical

components (V1x V1y )

V44m/s

V1y ?m/s

V1x ?m/s

Searching for y

Searching for x

An arrow is shot at 44m/s at an angle of 60

- Find the hang time
- find the horizontal distance the arrows travels.
- c) The maximum height of the arrow.

Horizontal Horizontal Horizontal Vertical Vertical Vertical

V K W V K W

a g -9.8

v1x 22 v1y 38.1

v2x v2y

dx dx dy dy

t t t

A football player kicks a ball at 44m/s at an

angle of 60.

- Find the hang time

Symmetry

A football player kicks a ball at 44m/s at an

angle of 60.

b) Find the horizontal distance

- A football player kicks a ball at 44m/s at an

angle of 60. Find the max height

Recall, vy0 at dy max

Practice Problems

- WS 7c
- 4
- WS 7d
- 1-3

Concept Questions

- A stone is thrown horizontally from a cliff.
- How would the x distance change if the stone was

thrown twice as fast?

- dx distance would double

Concept Questions

- How would the v2y change if the stone was thrown

twice as fast? - Vx and Vy are independent
- Since it was thrown horizontally, it would not

change.

Concept Questions

- How would the v2y change if the cliff was twice

as high?

The Partial Parabola

- Recall, this path has elevation and launch angle.
- The trajectory again has an apex.
- This is mathematically the most complex path.

Fire

Sample Partial Parabola Problem

- A cannon nestled in the side of a cliff (d1y

20m) fires a cannonball at 130m/s at a 30 angle.

Find v1x v1y Find y max Find t1 to apex Find t2

apex to ground (half parabola) Find total

time Find x max

Fire

Sample Partial Parabola Problem

- A cannon nestled in the side of a cliff (d1y

20m) fires a cannonball at 130m/s at a 40 angle.

Fire

Partial Parabola

Variable x y

a

v1

v2

d1

d2

t

- You are trying to win a prize by throwing an

apple into a basket on top of a pedestal. - The apple leaves your hand 1.00 m beneath the top

of the pedestal. - The apple flies 3.09 m horizontally before

landing in the bottom of the basket. - The apples maximum height was 1.26 m.
- What was the apples initial velocity (magnitude

and direction)?

Partial Parabola

- Find V1y knowing Dy max 1.26m

Dy (max) 1.26m

Partial Parabola

- Find the hang time knowing v1y and ending height

d2y1m

Partial Parabola

- Partial parabolas can be represented by two half

parabolas - Total hang time is the time of the ½ parabola on

the way up plus the time of the ½ parabola on the

way down.

Down

Up

The Partial Parabola

- If you look at this path carefully, you can see

two half parabolas, which simplifies things

considerably. - You still must consider the launch angle and the

components of the velocity when trying to solve.

1

2

Practice Problems

- WS 7c
- 1-6
- 8-9

Mortar Problems

V x y

a

v1

v2

d1

d2

t

- A mortar crew fires a projectile at an enemy

ammunitions storage facility that is protected by

a wall located on top of a 200.0 m high cliff. - The ammunition is located a horizontal distance

of 314.68 m from the mortars position. - The projectile passes directly over the wall at

its maximum height of 215.24 m. - What was the projectiles initial velocity

(magnitude and direction)?

Mortar Problems

End Chapter 7 Projectile motion

Kinematics in Dimensions

Two

Simple Harmonic Motion

- Simple Harmonic Motion Motion caused by a linear

restoring force that has a period independent of

amplitude. - Period The time required to repeat one complete

cycle - Amplitude Maximum displacement from equilibrium.

12

1

11

2

10

3

9

4

8

5

7

6

Simple Harmonic Motion

Periodic Motion is a form of Simple Harmonic

Motion

Restoring force

Restoring force

Equilibrium Position

Conservation of Energy

PE

KE

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Var Known Want

a 9.8m/s2

vyi

vyf

d1 44m

d2 0m

t t

- A stone is thrown horizontally from the top of a

44m high cliff at 15m/s. - a) How long does it take to reach the bottom?

- A stone is thrown horizontally from the top of a

44m high cliff at 15m/s. - a) How long does it take to reach the bottom?

- A stone is thrown horizontally from the top of a

44m high cliff at 15m/s. - b) How far from the base does it land?

Acceleration due to Gravity

Free Fall

- If you drop a book and a piece of paper which

will hit the floor first?

Acceleration due to Gravity

Simply replace a with g.

Acceleration due to Gravity

Gravity Time Instant speed Average speed Distance

10m/s2 0s 0m/s 0m/s 0m

10m/s2 1s 10m/s 5m/s 5m

10m/s2 2s 20m/s 10m/s 20m

10m/s2 3s 30m/s 15m/s 45m

10m/s2 4s 40m/s 20m/s 80m

10m/s2 5s 50m/s 25m/s ½ gt2

Graph shapes

- What geometric shape would that object have on a

position versus time graph after experiencing

acceleration?

- Given an initial velocity of 30m/s and an angle

of 35 - find v1x v1y

Searching for y

Searching for x

V130m/s

v1y

35o

v1x