Kinematics - Birdville Independent School District

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13 Νοε 2013 (πριν από 3 χρόνια και 7 μήνες)

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Kinematics

Velocity and Acceleration

Motion


Change in position of object in relation to
things that are considered stationary


Usually earth is considered stationary


Nothing is truly stationary (earth travels
108,000 km/hr orbiting sun)


All motion is relative: must be related to
other objects called your frame of reference


Distance and Displacement


Distance: how far object moves without
respect to direction, a scalar quantity


Displacement: change of position in a
particular direction; How far and in what
direction object is from original position, a
vector


Both use symbol
d

,and often
x

or
y

for 1
-
dim motion


The unit is the meter

Speed


The time rate of motion, the rate of change
of position
--
a scalar


Units of distance /time: m/s usually but can
be miles/hr, km/hr; Symbol is
v


Average speed = total distance/elapsed time


Instantaneous speed: rate of change of
position at any instant

Velocity


Speed in a particular direction, a vector; Unit same
as speed; Symbol
v


Must include a direction, using angle from known
reference points, compass headings, or just left &
right, + &
-
,up & down


Can be negative (going backwards)


Average velocity = total displacement / elapsed
time

Velocity


Instantaneous velocity: instantaneous
speed with current direction


Constant velocity means no change of
speed or direction


Often we are interested in only the speed
(we may know the direction) so speed
and velocity are sometimes used
interchangeably

Acceleration


Time rate of change of velocity; A
vector; Symbol
a

and units of m/s/s
usually shortened to m/s
2


Acceleration can be negative


Average acceleration = change in
velocity / elapsed time for the change


Galileo first to understand acceleration

1st Constant Accel. Equation


If acceleration is constant, instantaneous
acceleration always equals avg acceleration


Use definitions of avg velocity and accel to
calculate final velocity or distance


Since
a = (v
f

-

v
i
)/t

, then
v
f
= v
i

+ at


If
v
i
= 0

, then
v
f
= at


Use when distance not given or asked for

2nd Constant Acceleration
Equation


v
avg

= (v
f

+ v
i
)/
2 ; but also
v
avg

= d/t ;
so
(v
f

+ v
i
)/
2 =
d/t


Now using our first equation for
v
f

we can
get
(v
i

+ v
i

+ at)/
2 =
d/t



Solving for
d
:
d = v
i
t + 1/2 at
2


If
v
i

=
0,
d = 1/2 at
2


Use when final speed not given or asked for

3rd Constant Acceleration
Equation


Solve 1st equation for
t

and substitute into
2nd equation, expand squared quantity and
combine terms.


Get 2
ad = v
f

2
-

v
i
2
;
solve for
v
f

2


v
f

2
= v
i
2

+
2
ad


If
v
i

=
0,
v
f

2
=
2
ad


Use when time is not given or asked for

Graphing Motion:
d

vs
t


Plot time as independent variable


On position vs time graph, slope at any
value of
t

gives instantaneous velocity


If graph is linear, slope and
v
are constant



If graph is curved, slope and
v

are found
by drawing tangent line to curve and
finding its slope

Graphing Motion:
d

vs
t


Uniform motion (constant velocity)

Graphing Motion:
d

vs
t
(
x
vs

t
)


Accelerated motion (increasing velocity)

Graphing Motion:
v

vs
t


Slope of
v

vs
t

graph gives acceleration


If graph is linear, acceleration is
constant


If graph is curved, instantaneous
acceleration is found using slope of
tangent line at any point

Tangent Line

A line that just touches a curve at one point and gives the slope of the
curve at that point.

Velocity vs Time:
acceleration

Comparing Uniform and
Accelerated Motion Graphs


Uniform motion Accelerated Motion

Comparing Uniform and Accelerated
Motion Graphs



Uniform motion Accelerated Motion

Comparing Positive and Negative
Velocity


Speeding up and Slowing Down


Velocity vs Time Graphs:

Finding
Displacement


Displacement can be found from velocity graph by
finding the area between the line of the graph and
the time axis


Divide the area bounded by the graph line, the
horizontal axis and the initial and final times into
geometric sections (squares, rectangles, triangles,
trapezoids) and find the area


Area below the time axis is negative displacement

Area under (enclosed by) the
Velocity Graph

Area Enclosed by the Velocity
Graph


Divide complex areas into triangles and
rectangles

Area Enclosed by the
Acceleration Graph


If acceleration vs. time is plotted, area
between the graph line and the horizontal
(time) axis gives the change in velocity that
took place during the time interval

Free Fall


Common situation for constant
acceleration is free fall


Force of gravity causes falling bodies
to accelerate


Force varies slightly from place to
place but average acceleration is 9.80
m/s
2

designated by symbol
g



Often for simplicity or approximations,
g

= 10 m/s
2

is used

Free Fall


Distance increases with each
second of falling.


Object will fall 4.9 m (about 5
m) during the 1
st

second


Distance increases by 9.8m
(about 10 m) each second


Speed increases by 9.8 m/s
(about 10 m/s) for each
second of falling

Keeping Track of the Signs


If motion is only in one direction (usually
down), using positive and negative signs to
indicate direction is not necessary.


With up and down motion, up is considered
positive and down negative


g

must be negative (
-
9.80 m/s
2
) in these
situations along with downward displacements
and velocities

Air Resistance and Free Fall


If air drag is ignored, all objects fall at
the same rate


Air resistance slows rate of fall,
depending on object’s surface area,
shape, texture and density of air


For our purposes, air resistance is
negligible

Equations for Free Fall


Can use all constant acceleration equations
for free fall


Equations for vertical motion are written
with symbol
g

in place of
a

and
y

in place of
x

or
d


Be careful with positive and negative signs!

Constant Acceleration Equations


Horizontal Motion


Vertical Motion

t
a
v
v
i
f



2
2
1
)
(
t
a
t
v
x
i





x
a
v
v
i
f



2
2
2
t
g
v
v
i
f



2
2
1
)
(
t
g
t
v
y
i





y
g
v
v
i
f



2
2
2


1
2
i f
x v v t
  


1
2
i f
y v v t
  