Reminder: Free Fall
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
Free Fall
• Special case of motion with
constant acceleration,
acceleration of Earth’s
gravitational field g= 9.80 m/s2
• Important: as much as possible
try to keep a fixed convention
about what is the positive y
direction
• Recommend: positive y is “up”.
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
Free Fall Height after 1, 2, 3 .. s
• Acceleration of Earth’s
gravitational field g= 9.80 m/s
• Recall:
2
• Let us consider v = 0, y =0: start
i
i
from standing still at origin
t
1s
2s
3s
4s
y
– ½gs2
– 2gs2
– 4.5gs2
– 8gs2
y/y(1s)
1
4
9
16
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
Distance fallen in
th
n
second
An object is released from rest at t = 0 s. What distance
will it fall during the 4th second?
4
Distance fallen in
th
n
second
An object is released from rest at t = 0 s. What distance
will it fall during the 4th second?
t = 0s
1s
2s
3s
4s
5s
6s
...
4
Distance fallen in
th
n
second
An object is released from rest at t = 0 s. What distance
will it fall during the 4th second?
4th second, ti = 3s; tf = 4s
t = 0s
1s
2s
3s
4s
5s
6s
...
4
Experiment:
1:4:9:16 Free Fall demonstration
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
PHY 2053, Lecture 4:
Motion in a Plane
Axes, Unit Vectors and
Vector Components
1D person in a 2D world
1-Dim BFF
1D person in a 2D world
1-Dim BFF
1D person in a 2D world
1-Dim BFF
need lift
1D person in a 2D world
1-Dim BFF
need lift
whr R U
1D person in a 2D world
1-Dim BFF
need lift
whr R U
50 mi frm U
1D person in a 2D world
1-Dim BFF
need lift
whr R U
50 mi frm U
50 miles
1D person in a 2D world
1-Dim BFF
need lift
whr R U
50 mi frm U
wat
50 miles
Displacement in 2D
2-Dim BFF
need lift
whr R U
Displacement in 2D
2-Dim BFF
need lift
whr R U
50 mi NE of U
Displacement in 2D
2-Dim BFF
50 miles
need lift
whr R U
50 mi NE of U
Displacement in 2D
2-Dim BFF
50 miles
need lift
whr R U
50 mi NE of U
Displacement in 2D
2-Dim BFF
50 miles
need lift
whr R U
50 mi NE of U
Displacement in 2D
2-Dim BFF
50 miles
need lift
whr R U
50 mi NE of U
Ok drivin
Displacement in 2D
2-Dim BFF
y
50 miles
need lift
whr R U
x
50 mi NE of U
Ok drivin
How do we define a position in 2D? We need:
• a point of reference (“origin”)
• a map-building convention (“coordinate system”)
• We just found we need both magnitude and direction
to describe a position in 2 dimensions
• Call that object a “vector”:
• How do we add / subtract vectors with that definition?
PHY2053, Lecture 4, Motion in a Plane
10
North
Vector Algebra
East
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
34
Vector Algebra
North
4 km
3 km
2 km
1 km
1 km
2 km
3 km
4 km
5 km
6 km
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
7 km
East
34
Vector Algebra
North
4 km
3 km
2 km
First, we hiked
4 km East and
1 km North
1 km
1 km
2 km
3 km
4 km
5 km
6 km
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
7 km
East
34
Vector Algebra
North
4 km
Then, we hiked 1
km West and 3
km North
3 km
2 km
First, we hiked
4 km East and
1 km North
1 km
1 km
2 km
3 km
4 km
5 km
6 km
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
7 km
East
34
Vector Algebra
North
4 km
Then, we hiked 1
km West and 3
km North
3 km
2 km
First, we hiked
4 km East and
1 km North
1 km
1 km
2 km
3 km
4 km
East
North
4 km
1 km
-1 km
3 km
3 km
4 km
5 km
6 km
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
7 km
East
34
Vector Algebra
North
4 km
3 km
Then, we hiked 1
km West and 3
km North
Ok, to find you I
have to hike 3 km
East and 4 km
North.
2 km
First, we hiked
4 km East and
1 km North
1 km
1 km
2 km
3 km
4 km
East
North
4 km
1 km
-1 km
3 km
3 km
4 km
5 km
6 km
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
7 km
East
34
Vector Algebra
4 km
3 km
Then, we hiked 1
km West and 3
km North
Ok, to find you I
have to hike 3 km
East and 4 km
North.
2 km
First, we hiked
4 km East and
1 km North
1 km
1 km
2 km
3 km
4 km
East
North
4 km
1 km
-1 km
3 km
3 km
4 km
5 km
6 km
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
7 km
34
Vector Algebra
4 km
3 km
2 km
1 km
1 km
2 km
3 km
4 km
5 km
6 km
UF PHY2053, Lecture 4: Motion in One Dimension and On A Plane
7 km
34
Vector reminder
• vector addition
• vector scaling
• vector subtraction
• vector magnitude
PHY2053, Lecture 4, Motion in a Plane
12
Vector reminder
• vector addition
• vector scaling
• vector subtraction
• vector magnitude
PHY2053, Lecture 4, Motion in a Plane
12
Vector reminder
• vector addition
• vector scaling
• vector subtraction
• vector magnitude
PHY2053, Lecture 4, Motion in a Plane
12
Vector reminder
• vector addition
• vector scaling
• vector subtraction
• vector magnitude
PHY2053, Lecture 4, Motion in a Plane
12
Vector reminder
• vector addition
• vector scaling
• vector subtraction
p
2
2
~
vector
magnitude
|•A| = ax + ay
PHY2053, Lecture 4, Motion in a Plane
12
Vector Direction
13
Vector Direction
• Define coordinate system
13
Vector Direction
• Define coordinate system
• The vector direction is its angle with
respect to the “x” axis
13
Vector Direction
• Define coordinate system
• The vector direction is its angle with
respect to the “x” axis
• Follow “right hand rule” for sign
positive angle
13
Vector Direction
• Define coordinate system
• The vector direction is its angle with
respect to the “x” axis
• Follow “right hand rule” for sign
positive angle
negative angle
13
Vector Direction
• Define coordinate system
• The vector direction is its angle with
respect to the “x” axis
• Follow “right hand rule” for sign
13
Vector Direction
• Define coordinate system
• The vector direction is its angle with
respect to the “x” axis
• Follow “right hand rule” for sign
13
Vector Direction
• Define coordinate system
• The vector direction is its angle with
respect to the “x” axis
• Follow “right hand rule” for sign
⇣ ⌘
✓ = tan
✓ = sin
✓ = cos
1
1
1
y
x!
y
~
|A|
x
~
|A|
!
13
Adding Vectors, Graphically
PHY2053, Lecture 4, Motion in a Plane
14
Adding Vectors, Graphically
A
PHY2053, Lecture 4, Motion in a Plane
14
Adding Vectors, Graphically
B
A
PHY2053, Lecture 4, Motion in a Plane
14
Adding Vectors, Graphically
B
A
PHY2053, Lecture 4, Motion in a Plane
14
Adding Vectors, Graphically
B
A+B
A
PHY2053, Lecture 4, Motion in a Plane
14
Subtracting
Vectors
PHY2053, Lecture 4, Motion in a Plane
15
Subtracting
Vectors
PHY2053, Lecture 4, Motion in a Plane
15
Subtracting
Vectors
PHY2053, Lecture 4, Motion in a Plane
15
Subtracting
Vectors
PHY2053, Lecture 4, Motion in a Plane
15
Subtracting
Vectors
PHY2053, Lecture 4, Motion in a Plane
15
Subtracting
Vectors
PHY2053, Lecture 4, Motion in a Plane
15
Subtracting
Vectors
PHY2053, Lecture 4, Motion in a Plane
15
Subtracting
Vectors
PHY2053, Lecture 4, Motion in a Plane
15
Subtracting
Vectors
PHY2053, Lecture 4, Motion in a Plane
15
H-ITT Problem #1, 3.5 min
PHY2053, Lecture 4, Motion in a Plane
16
Velocity in Vector Form
PHY2053, Lecture 4, Motion in a Plane
17
Velocity in Vector Form
• Reminder: “Average velocity is the change in
position divided by the time interval in which it took
place”
~
rf ~
ri
~
r
~
v av ⌘
tf
PHY2053, Lecture 4, Motion in a Plane
ti
=
t
17
Velocity in Vector Form
• Reminder: “Average velocity is the change in
position divided by the time interval in which it took
place”
~
rf ~
ri
~
r
~
v av ⌘
tf
ti
=
t
• Reminder: “Instantaneous velocity is the limit of the
average velocity as the time interval becomes infinitesimally
short.”
~
rf
~
v ⌘ lim
tf !ti tf
PHY2053, Lecture 4, Motion in a Plane
~
ri
= lim
t!0
ti
~
r
t
17
Velocity, Graphically..
Trajectory
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
ri
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
ri
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Velocity, Graphically..
Trajectory
rf − ri
ri
vav
rf
Origin
~
v av
PHY2053, Lecture 4, Motion in a Plane
~
rf
⌘
tf
~
ri
=
ti
~
r
t
18
Acceleration in Vector Form
PHY2053, Lecture 4, Motion in a Plane
19
Acceleration in Vector Form
• Reminder: “Average acceleration is the change in
velocity divided by the time interval in which it took
place”
~
vf ~
vi
~
v
~
aav ⌘
tf
PHY2053, Lecture 4, Motion in a Plane
ti
=
t
19
Acceleration in Vector Form
• Reminder: “Average acceleration is the change in
velocity divided by the time interval in which it took
place”
~
vf ~
vi
~
v
~
aav ⌘
tf
ti
=
t
• Reminder: “Instantaneous acceleration is the limit of the
average acceleration as the time interval becomes
infinitesimally short.”
~
vf
~
a ⌘ lim
tf !ti tf
PHY2053, Lecture 4, Motion in a Plane
~
vi
= lim
t!0
ti
~
v
t
19
Acceleration, Graphically..
Trajectory
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
Trajectory
ri
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
Trajectory
ri
rf
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
Trajectory
ri
rm
rf
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
rm − ri
ri
Trajectory
rm
rf
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
rm − ri
ri
Trajectory
rf − rm
rm
rf
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
Trajectory
vi
ri
rf − rm
rm
rf
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
Trajectory
vi
ri
rm
vf
rf
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
vi
vi
ri
rm
vf
Trajectory
vf
rf
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
vi
vi
ri
Trajectory
vf
vf − vi
rm
vf
rf
Origin
PHY2053, Lecture 4, Motion in a Plane
20
Acceleration, Graphically..
vi
vi
ri
vf
vf − vi
rm
vf
a
rf
Origin
Trajectory
~
aav
PHY2053, Lecture 4, Motion in a Plane
~
vf
⌘
tf
~
vi
=
ti
~
v
t
20
H-ITT Problem #2, 3 min
PHY2053, Lecture 4, Motion in a Plane
21