# Kinematics - Birdville Independent School District

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13 Νοε 2013 (πριν από 4 χρόνια και 6 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
f

2
-

v
i
2
;
solve for
v
f

2

v
f

2
= v
i
2

+
2

If
v
i

=
0,
v
f

2
=
2

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

Can use all constant acceleration equations

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
  