Physics I
95.141
LECTURE 5
9/20/10
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Outline
• Review of Lecture 4
• Projectile Motion
• What do we know?
–
–
–
–
–
–
Units
Kinematic equations
Freely falling objects
Vectors
Kinematics + Vectors = Vector Kinematics
Relative motion
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Exam Prep Problem
• An object starts from
If the acceleration of the
 rest atˆ the ˆorigin.
object is given by: a (t )  3i  2 j  kˆ
• 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?
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Exam Prep Problem
• An object starts from
If the acceleration of the
 rest atˆ the ˆorigin.
object is given by: a (t )  3i  2 j  kˆ
• A) (10pts) Give the velocity and displacement of the object, as a
function of time.
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Exam Prep Problem
• An object starts from
If the acceleration of the
 rest atˆ the ˆorigin.
object is given by: a (t )  3i  2 j  kˆ
• B) (5pts) What is the object’s velocity and speed at 10s?
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Exam Prep Problem
• An object starts from
If the acceleration of the
 rest atˆ the ˆorigin.
object is given by: a (t )  3i  2 j  kˆ
• C) (5 pts) What is the object’s displacement at 10s?
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Exam Prep Problem
• An object starts from
If the acceleration of the
 rest atˆ the ˆorigin.
object is given by: a (t )  3i  2 j  kˆ
• D) (5 pts) What is the average velocity of the object for the first
10 seconds of motion?
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Projectile Motion (displacement)
• Projectile motion is a special case of motion with
constant acceleration: the acceleration due to

gravity
ˆ
ˆ
m
a  gj  9.8
s2
j
• Here, the acceleration is in only one direction!
• The equations of motion become:

1 2ˆ

ˆ
r (t )   xo  v ox t  i   yo  v oy t  gt  j
2



1

2ˆ
ˆ
r (t )   xo  v ox t  i   yo  v oy t  9.8t  j
2


95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Projectile Motion (Equations Of Motion)
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Projectile Motion (velocity)
• We can always find the expression for velocity
by differentiating the expression for
displacement with respect to time.

g 2ˆ

ˆ
r (t )   xo  v ox t  i   yo  v oy t  t  j
2 

95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Projectile Motion (acceleration)
• We can always find the expression for
acceleration by differentiating the expression for
velocity with respect to time.

v (t )  vox iˆ  voy  9.8t  ˆj
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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
y
Vyo
x
Vxo
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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)
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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?
y
x
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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
2
• What does this mean in numbers?
y
x
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
R
v o sin 2 o
g
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?
y
x
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Projectile Motion
• For a typical projectile motion problem, we can
think about the object motion in component form.
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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?
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
The Speed Bus

• OK, so we know: vo
1) DRAW DIAGRAM!!
2) Determine knowns
3) Pick Equations
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Speed Bus with Magic Launch

• OK, so we know new vo
1) DRAW DIAGRAM!!
2) Determine knowns
3) Pick Equations
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Does it make it?
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Helicopter, Part (a)
• Divide equations into x and y
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
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
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Helicopter, Part (b)
• Divide equations into x and y
95.141, F2010, Lecture 5
Department of Physics and Applied Physics
Now We Know
• Projectile Motion
– Motion in component form
– Problem solving approach
95.141, F2010, Lecture 5
Department of Physics and Applied Physics