# 01.Kinematics

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14 Νοε 2013 (πριν από 4 χρόνια και 7 μήνες)

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Kinematics

Study of an object’s motion without
regard to the forces causing the motion.

Kinematics asks three questions at a
particular instant in time

Where is the object?

Position:

distance and direction (“displacement”) from a
given reference point. (Ex: 4 meters
E
ast of the wall)

How fast is the object moving and in what direction?

Instantaneous Velocity:
The rate at which the object
changes position. (Ex: 4 meters per second East.)

How fast is the object speeding up, slowing down or
changing direction?

Acceleration:
The rate at which velocity changes. (Ex: 4
meters per second each second East).

Two Types of Quantities in Physics

Scalars
: Have “
magnitude
” (amount with units)
only.

Ex:
Speed
: 4 meters per second

Vectors:

Have magnitude
and
direction.

Ex:
Velocity
: 4 meters per second East

Identify the following as a vector or scalar:

A mass: 6 kilograms (6 kg)

A distance: 6 meters (4m)

A displacement: 6 meters (6m) East

A position: 6 meters (6m) East of the wall.

Table of Kinematic Quantities

QUESTION

SCALAR

VECTOR

S.I.
UNIT

When?

Where?

How Fast?

Changing
Velocity?

Table of Kinematic Quantities

QUESTION

SCALAR

VECTOR

(vector symbols are
in
boldface
)

S.I.
UNIT

When?

Time (t)

------------

Second
(s)

Where?

Distance (d)

Position (
x
,
y
)

Displacement (
d
)

Meter
(m)

How Fast?

Instantaneous
Speed (v)

Instantaneous
Velocity (
v
)

m/s

Changing
Velocity?

Acceleration (a)

Acceleration (
a
)

m/s/s,
or m/s
2

Average Speed and Velocity

(Always applies to a TRIP)

Average
Speed

= (distance of trip)/(time of trip)

V
AV

= d/t

Average
Velocity

= (displacement of trip)/(time of trip)

V
AV

=
d
/t

(Note: The
boldface
type means the quantity
ia

a
vector
)

How does average walking pace
vary from person to person?

Each person walks
naturally
along the route
specified: d = ______ ft.

The time for the walker is measured by his/her
partner with a stopwatch.

The average speed is then: v
AV

= d/t

Convert from feet per second to miles per hour.

If the trip’s endpoint is the same as the starting
point, what is each person’s average
velocity
?

How do we convert from feet
per second to miles per hour?

Three Cases of Kinematic Motion

Definition of Acceleration:
a

=
Δ
v
/
Δ
t

In each case we are given: Initial velocity (
v
i
) and
acceleration (a)

We’ll look at three cases:

Case 1: Constant velocity Motion: v
i
≠ 0, a = 0

Case 2: Accelerating from Rest:
v
i
= 0, a ≠ 0

Case 3: Accelerating from Nonzero initial velocity:
v
i
≠ 0, a ≠ 0

In each case, we want to know:

Velocity at a particular time (t). How Fast?

Distance traveled after a particular time (t). How Far?

We plot v vs. t by knowing
v
i

and a.

Case 1: Constant Velocity Motion

Car moving at constant
velocity:
v
i
≠ 0, a = 0

Ex: v
i

= 4 m/s,

Plot v vs. t

What does the slope of
the v vs. t graph mean?

What does the area under
the v vs. t graph mean?

Plot d vs. t

t (s)

v (m/s)

d (m)

0

1

2

3

4

5

Case 2: Accelerating from Rest:
v
i
=0, a ≠ 0

A car accelerates from rest
at a constant rate of 2 m/s
2

Plot v vs. t

What does the slope of the v
vs. t graph mean?

What does the area under
the v vs. t graph mean?

Can we find v and d using
algebra?

Plot d vs. t

t (s)

v (m/s)

d (m)

0

V
i
=0

1

2

3

4

5

How do free falling objects differ
in acceleration due to gravity?

Measure the distance (d) from the ceiling to
the floor.

Drop various objects and time how long it
takes each one to hit the floor (t). Describe
each object:
Ping
-
Pong, wood, copper tube,

Calculate the acceleration d = ½ at
2

or: a = 2d/t
2

for each ball.

Results of Free Falling Balls

(
d = 2.765m)

Accepted value: g = 9.81 m/s
2

= acceleration due to gravity.

Object

Mass (g)

Acceleration (a)
(m/s
2
)

Ping
-
Pong

2.3

Wood

2.5

Rubber

8.9

Copper Tube

12.1

Small Steel

28.2

Brass Mass

50.0

93.5

Results of Free Falling Balls

(
d = 2.765m)

Accepted value: g = 9.81 m/s
2

= acceleration due to gravity.

Object

Mass (g)

Acceleration (a)
(m/s
2
)

Ping
-
Pong

2.3

Wood

2.5

Rubber

8.9

Copper Tube

12.1

Small Steel

28.2

Brass Mass

50.0

93.5

How can we estimate error due to our
reaction time?

How High Is This Cliff?

1.
You’re at the top of a high
cliff, overlooking a river.
How can you find out its
height?

2.
It takes
10 seconds
for a
dropped rock to hit the
river below. How high is
the cliff?

3.
How
Fast

is the rock
traveling after 3.5
seconds?

4.
How
Far

has the rock
traveled after 3.5 seconds?

Case 3: Accelerating from an initial
velocity: v
i
≠ 0, a ≠ 0

How can we find equations for v and d as
“functions” of time (t)?

From definition of acceleration, we get:

v = v
i

+ a (
Δ
t)

From area under v vs. t graph, we get:

d = v
i

(
Δ
t) + ½ a (
Δ
t)
2

Case 3: Accelerating from an
initial velocity: v
i
≠ 0, a ≠ 0

A car accelerates from v
i
= 4 m/s at a constant
rate of 2 m/s
2
.
Fill out the following table and
plot

v vs. t and d vs. t

v = v
i

+ a (
Δ
t)

d = v
i

(
Δ
t) + ½ a (
Δ
t)
2

t (s)

v (m/s)

d (m)

0

V
i
=4

1

2

3

4

5

Summarizing Kinematic Cases

Δ
t = time interval

v
i
= velocity at start of time interval

v = velocity at end of time interval

d = distance traveled at end of time interval

Case 1: Constant velocity: v
i
≠ 0; a = 0

v = v
i

, d = v
i
(
Δ
t)

Case 2: Accelerating from Rest: v
i
= 0; a ≠ 0

v = a (
Δ
t) , d = ½ a (
Δ
t)
2

Case 3: Accelerating from an initial velocity
v
i
≠ 0; a ≠ 0

v = v
i

+ a (
Δ
t)
(“Kinematic Equation 1”)

d = v
i

(
Δ
t) + ½ a (
Δ
t)
2
(“Kinematic Equation 2”)

Summarizing Kinematic Cases

(setting initial time at: t
0

=0)

Case 1: Constant velocity: v
i
≠ 0; a = 0

v = v
i

, d = v
i

t

Case 2: Accelerating from Rest: v
i
= 0; a ≠ 0

v = a t , d = ½ a t
2

Case 3: Accelerating from an initial velocity
v
i
≠ 0; a ≠ 0

v = v
i

+ a t
(“Kinematic Equation 1”)

d = v
i

t + ½ a t
2
(“Kinematic Equation 2”)

Summarizing Kinematic Cases

General Time Interval

Case 1: Constant velocity: v
0
≠ 0; a = 0

v = v
i

, d = v
i
(
Δ
t)

Case 2: Accelerating from
Rest: v
i
= 0; a ≠ 0

v = a (
Δ
t) , d = ½ a (
Δ
t)
2

Case 3: Accelerating from an
initial velocity v
i
≠ 0; a ≠ 0

v = v
i

+ a (
Δ
t)
(“Kinematic Equation 1”)

d = v
i

(
Δ
t) + ½ a (
Δ
t)
2
(“Kinematic Equation 2”)

Time interval starting at t
0
= 0

Case 1: Constant velocity:
v
i
≠ 0; a = 0

v = v
i

, d = v
i

t

Case 2: Accelerating from
Rest: v
i
= 0; a ≠ 0

v = a t , d = ½ a t
2

Case 3: Accelerating from an
initial velocity v
i
≠ 0; a ≠ 0

v = v
i

+ a t
(“Kinematic Equation 1”)

d = v
i

t + ½ a t
2
(“Kinematic Equation 2”)

Relative Motion

Person A stands on the sidewalk. Given:

A sees B drive by at 40 mi/h East.

A sees C drive by at 60 mi/h East.

Questions:

According to B

What is A’s velocity?

What is C’s velocity?

According to C

What is A’s velocity?

What is B’s velocity?

Suppose B throws a ball sideways out the window.

How does B see the ball move?

How does A see the ball move?