Motion in One Dimension

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14 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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Chapter 2

Motion in One Dimension

(Kinematics)

2.1
Displacement and Velocity


Distance

is a measure of the total
motion of
an object (how far it has traveled)


Displacement

is
the net distance and
direction an object has traveled (straight
line distance and direction from starting to
ending point)


o
What’s your
distance when you run two laps around a
400 m track?

o
What is your displacement?

o
How
are they different?

What is the gecko’s displacement?


∆x = x
f

-

x
i


∆x = 85 cm


25 cm


∆x = 60 cm




Displacement continued


Displacement is not always equal to the
distance traveled (if you move back and forth)


Displacement can be
positive

or
negative
.

Speed


We use the variable
speed

to describe how
quickly something moves.


Saying a race car, runner, or plane is “fast” is
not enough to accurately describe its speed
scientifically.


Speed

v =
d


t

Distance traveled (m)

Time taken (sec)

Speed (m/sec)

Calculate speed


A bird flies 50 meters in 7.5 seconds.


Calculate the speed of the bird in m/sec.


Calculate distance


How far do you go if you drive for 2 hours at a
speed of 100 kilometers per hour?

Calculate time and distance


A space shuttle is traveling at a
speed of 7,700 m/sec.


How far does the shuttle travel
in kilometers in one hour?


At an altitude of 300 kilometers,
the circumference of the
shuttle’s orbit is 42 million
meters.


How long does it take the
shuttle to go around the Earth
one time?

Calculating Speed


Why the letter
v
is used for speed...

o
We use the letter
v
to represent speed in a formula
.

o
We already used the letter s for seconds


Speed

is a single measurement that tells how
fast you are going, like 100 km/h.


Velocity

means you know both your speed
and your direction.

Instantaneous Speed and Velocity


Instantaneous speed
is
distance
÷

time interval

o
when the time interval
is
made
very small








t
x
v
inst




Average Speed


Average Speed
is the TOTAL DISTANCE divided by the
TOTAL TIME



o
When
is Average Speed = Instantaneous speed
?

tot
tot
av
T
D
v

Velocity


Average velocity is
displacement

divided by
the time interval


2.2 Acceleration


Acceleration measures the rate of change of
velocity


When an object speeds up or slows down it is
accelerating.


When an object is speeding up its acceleration
is
positive

and when it is slowing down its
acceleration is
negative


Acceleration movie

Constant Acceleration


Constant acceleration is sometimes called
uniform
acceleration
.


A ball rolling down a straight ramp has constant
acceleration.

constant acceleration

increasing speed

Acceleration
and
Speed


An object can have acceleration, but no
speed.


Consider a ball rolling up a ramp.


As the ball slows down, eventually its
speed
becomes
zero.

constant negative


acceleration

decreasing
speed

Speed
vs

Velocity


We frequently use the terms speed and
velocity interchangeably


But remember that velocity is speed in a
particular direction


So speed is always a
positive

number


Whereas velocity can
be a
negative

number if
you are going backwards.

Slope
and Acceleration


Use
slope

to recognize when
there is acceleration in
speed vs
.
time graphs.

o
Level sections
(A)

on the graph
show an acceleration of zero.

o
The highest acceleration
(B)

is the
steepest slope on the graph.

o
Sections that slope down
(C)

show
negative acceleration (slowing
down).

Slope
of a graph


The
slope

of a graph is equal
to the ratio of
rise
to
run
.


The
rise

is the amount the
speed changes.


The
run

is the amount the
time changes.

Acceleration
and slope


Acceleration

is the change in speed over the change in time
.


The slope of a speed
vs

time graph is acceleration.

Calculating Acceleration



The formula for acceleration can be expressed
several ways


𝑎
=

𝑣





=




𝑣
𝑓

𝑣
𝑖

𝑓


𝑖



=



𝑐ℎ𝑎𝑔𝑒

𝑖

𝑣𝑒 𝑐𝑖
𝑐ℎ𝑎𝑔𝑒

𝑖

𝑖𝑒


But we often write it as:
𝑎
=

𝑣
𝑓

𝑣
𝑖





Acceleration units


Acceleration is a unit of distance divided by a
unit of time squared.


In physics we generally use the unit:
m/s
2



(meters per second per second)



A biker goes from a speed of 0.0 m/s to a final
speed of 25.0 m/s in 10 seconds. What is the
acceleration of the bicycle?


𝑎
=

𝑣
𝑓

𝑣
𝑖





𝑎
=
25






0

10





𝑎
=
2
.
5


/

2

Kinematics Equations


You should now know how to calculate
average velocity and average acceleration
using the equations for velocity and
acceleration.


Next we will combine them so that you can
solve any problem related to objects that are
moving at a constant velocity or accelerating.


We need 3 more equations to do this.

Finding displacement (∆x) when you
know initial and final velocity and time



The first equation allows you to find
displacement when you know the initial and
final velocity of an object and the time that it
accelerated.



=
1
2
(
𝑣
𝑖
+
𝑣
𝑓
)




Question: A sports car accelerates from rest to a
velocity of 25 m/s in
5.0
seconds. How far did it travel
?





=
1
2
(
𝑣
𝑖
+
𝑣
𝑓
)






=
1
2
0
.
0


/

+
25


/

5
.
0

𝑒𝑐





=
𝟔

𝒎


Finding displacement (∆x) when you know
initial velocity, acceleration, and time.



The second equation allows you to find displacement
when you know initial velocity, acceleration, and
time (but not final velocity).





=

𝑣
𝑖
+
1
/
2𝑎

2



You used this equation to determine how high your
tennis ball went.

Question: A car traveling at 30 m/s brakes for 3.0
seconds at an acceleration of
-
2.0 m/s
2
. How far
did it travel during this time?





=

𝑣
𝑖
+
1
/
2𝑎

2





=
30


/

+
1
/
2
(

2
.
0


/

2
)
(
3
2
𝑒𝑐
)





=


𝒎

Finding final velocity when you know initial
velocity, acceleration, and time


The third equation is used to final velocity
when you know initial velocity, acceleration,
and time



𝑣
𝑓
=
𝑣
𝑖
+
𝑎



Question: A jet plane accelerates from rest down a runway
for 12 seconds at an acceleration of 15 m/s2. What is it’s
final velocity?



𝑣
𝑓
=
𝑣
𝑖
+
𝑎





𝑣
𝑓
=
0


/

+
(
15


/

2
)
(
12

sec

)



𝑣
𝑓
=
𝟖

𝒎
/
𝒔


Using the Equations of Kinematics


Choose a positive direction


Draw a picture


Make a chart of initial conditions


Choose an equation to solve for the
unknown


Solve and check answer for reasonability

Kinematics Example
#1



A car accelerates from a stop light for
20.0 s with an acceleration of 5.0 mph/s
2
.
Find its velocity at the end of this time.
How far has it traveled?

Kinematics Example
#1

V = 100 mph

X = 0.278 miles



Kinematics Example
#2


A 747 airliner needs to achieve a
speed of 360 km/h to take off. If
acceleration is uniform and the
runway is 1.8 km long, what
acceleration is needed?

Kinematics Example
#2

10 km/h/s

or 2.78 m/s
2


Kinematics Example
#3



A penny is dropped from a
building rooftop and takes 6.0
s to hit the ground. What
velocity does it hit the ground
with? How tall is the building
in meters? in feet? How
would this change if you
accounted for the speed of
sound?

Kinematics Example
#3

H=176m = 579ft

V=58.8 m/s



Example #4 A
High Speed Chase



In a high speed chase, a police car takes off
after a getaway car going at its maximum
speed of 90 mph. The police car accelerates
at a rate of 10 mph/s to a maximum speed of
120 mph. How long does it take to catch the
bad guys?

Free Fall


An object is said to be in “free
fall” if it experiences
only

a force
of gravity.


o
We abbreviate the
accel
. of gravity as “
g


o
g=
9.81m/s
2

near the Earth’s surface

What goes up…


A ball is tossed in the air..

o
When is its velocity 0?

o
When is its acceleration positive?
Negative?

Kinematics Example #
4


A ball is thrown straight up in the air with an
initial velocity of 25.0 m/s. How long will it
take for the ball to reach its starting point?
What will the ball’s velocity be when it returns
to its starting point?

Kinematics Example # 6

t = 5.08 s

v
f

= 25.16 m/s



The End…