Department of Physics and Applied Physics95.141, F2010, Lecture 5

Physics I95.141

LECTURE 59/20/10

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Outline

• Review of Lecture 4• Projectile Motion

• What do we know?– Units

– Kinematic equations

– Freely falling objects

– Vectors

– Kinematics + Vectors = Vector Kinematics

– Relative motion

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Exam Prep Problem• An object starts from rest at the origin. If the acceleration of the

object is given by:• A) (10pts) Give the velocity and displacement of the object, as a

function of time.• B) (5pts) What is the object’s velocity and speed at 10s?• C) (5 pts) What is the object’s displacement at 10s?• D) (5 pts) What is the average velocity of the object for the first

10 seconds of motion?

kjita ˆˆ2ˆ3)(

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Exam Prep Problem• An object starts from rest at the origin. If the acceleration of the

object is given by:• A) (10pts) Give the velocity and displacement of the object, as a

function of time.

kjita ˆˆ2ˆ3)(

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Exam Prep Problem• An object starts from rest at the origin. If the acceleration of the

object is given by:• B) (5pts) What is the object’s velocity and speed at 10s?

kjita ˆˆ2ˆ3)(

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Exam Prep Problem• An object starts from rest at the origin. If the acceleration of the

object is given by:• C) (5 pts) What is the object’s displacement at 10s?

kjita ˆˆ2ˆ3)(

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Exam Prep Problem• An object starts from rest at the origin. If the acceleration of the

object is given by:• D) (5 pts) What is the average velocity of the object for the first

10 seconds of motion?

kjita ˆˆ2ˆ3)(

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Projectile Motion (displacement)

• Projectile motion is a special case of motion with constant acceleration: the acceleration due to gravity jjga

sm ˆ8.9ˆ

2

• Here, the acceleration is in only one direction!• The equations of motion become:

jttvyitvxtr

jgttvyitvxtr

oyooxo

oyooxo

ˆ8.92

1ˆ)(

ˆ2

1ˆ)(

2

2

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Projectile Motion (Equations Of Motion)

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Projectile Motion (velocity)

• We can always find the expression for velocity by differentiating the expression for displacement with respect to time.

jtg

tvyitvxtr oyooxoˆ

2ˆ)( 2

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Projectile Motion (acceleration)

• We can always find the expression for acceleration by differentiating the expression for velocity with respect to time.

jtvivtv oyoxˆ8.9ˆ)(

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Projectile Motion• Problem Solving Strategy

– Draw a diagram, choose coordinate system

– Split into x, y components of motion

– Think about what problem is actually asking!

– List unknowns and knowns

– Apply relevant equations and solve

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example

• Say I hit a golf ball with initial velocity vo at an angle of θº.– Find equations of motion– Find ball height as a function of lateral position (y(x))– Find the Range of the ball (assuming ground is flat)– The time of flight

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example Problem• Say I hit a golf ball with initial velocity vo at an angle of θº.

– A) Find equations of motion• Draw diagram and choose coordinate system

• Fill in knowns

Vxo

Vyo x

y

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example Problem• Say I hit a golf ball with initial velocity vo at an angle of θº.

– B) Find y(x)• Write out equations

• Solve for y(x)

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example Problem• Say I hit a golf ball with initial velocity vo at an angle of θº.

– C) Find Range (distance ball travels before hitting ground)• What does this mean in numbers?

x

y

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example Problem (Extra)• Say I hit a golf ball with initial velocity vo at an angle of θº.

– C+) Find the θ for maximum Range• What does this mean in numbers?

x

yg

vR oo 2sin2

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example Problem• Say I hit a golf ball with initial velocity vo at an angle of θº.

– D) Find time of flight (time ball travels before hitting ground)• What does this mean in numbers?

x

y

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Projectile Motion

• For a typical projectile motion problem, we can think about the object motion in component form.

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example Problem• A punter, on average, can give the football an initial velocity of

27m/s. The Cowboy’s new $1.2 Billion stadium has a scoreboard 90ft (27.5m) off the ground. What is the minimum angle required for an average punt to hit the scoreboard?– Find initial y-velocity required to hit scoreboard

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example Problem• A punter, on average, can give the football an initial velocity of

30m/s. The Cowboy’s new $1.2 Billion stadium has a scoreboard 90ft (27.5m) off the ground. What is the minimum angle required for an average punt to hit the scoreboard?– What is angle?

Department of Physics and Applied Physics95.141, F2010, Lecture 5

The Speed Bus

• OK, so we know: ov

1) DRAW DIAGRAM!!2) Determine knowns3) Pick Equations

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Speed Bus with Magic Launch

• OK, so we know new1) DRAW DIAGRAM!!2) Determine knowns3) Pick Equations

ov

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Does it make it?

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example (Rescue Helicopter)

• Helicopter wants to drop supplies on mountain top 200m below. Helicopter flying horizontally at 70m/s– A) How far in advance (horizontal distance) should the package

be dropped? • Draw diagram, choose

coordinate system• Knowns and unknowns

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Helicopter, Part (a)

• Divide equations into x and y

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Example (Rescue Helicopter)

• Helicopter wants to drop supplies on mountain top 200m below, 400m in advance. Helicopter flying horizontally at 70m/s– B) What vertical velocity should the package be given?

• Draw diagram, choose coordinate system, time interval

• Write out equations

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Helicopter, Part (b)

• Divide equations into x and y

Department of Physics and Applied Physics95.141, F2010, Lecture 5

Now We Know

• Projectile Motion– Motion in component form– Problem solving approach