Copyright Sautter 2003
Measuring Motion
•
The study kinematics requires the measurement of
three properties of motion.
•
(1) displacement
–
the straight line distance between
two points (a vector quantity)
•
(2) velocity
–
the change in displacement with respect
to time (a vector quantity)
•
(3) acceleration
–
the change in velocity with respect to
time (a vector quantity)
•
The term distance like displacement, refers to the
change in position between two points, but not in a
straight line. Distance is a scalar quantity. Speed refers
to change in position with respect to time but unlike
velocity, does not require straight line motion. Speed is
a scalar quantity.
Lake Tranquility
A x
x B
Distance traveled
from A to B
Displacement
from A to B
Speed = Distance from A to B/ time
Velocity = Displacement from A to B/ time
VELOCITY & ACCELERATION
•
OBJECTS IN MOTION MAY MOVE AT CONSTANT
VELOCITY (COVERING EQUAL DISPLACEMENTS IN
EQUAL TIMES) OR BE ACCELERATED (COVER
INCREASING OR DECREASING DISPLACEMENTS IN
EQUAL TIMES).
•
VELOCITY MEASUREMENTS MAY BE OF TWO TYPES,
AVERAGE VELOCITY (VELOCITY OVER A LARGE
INTERVAL TIME) OR INSTANTANEOUS VELOCITY
(VELOCITY OVER A VERY SHORT INTERVAL OF
TIME).
•
ACCELERATION MAY BE UNIFORM OR NON
UNIFORM. UNIFORM OR CONSTANT ACCELERATION
REQUIRES THAT THE VELOCITY INCREASE OR
DECREASE AT A CONSTANT RATE WHILE NON
UNIFORM ACCELERATION DISPLAYS NO REGULAR
PATTERN OF CHANGE.
1 sec
2 sec
3sec
4sec
5 sec
EQUAL DISPLACEMENTS IN EQUAL TIMES
1 sec
2 sec
3sec
4sec
REGULARLY INCREASING DISPLACEMENTS IN EQUAL TIMES
CLICK
HERE
S
D
I
S
P
L
A
C
E
M
E
N
T
time
t
S
t
S
t
偏卉呉噅
ACCELERATION
Equal time
intervals result
in increasingly
larger displacements
Average velocity
between
t
1
and t
2
Is the slope of the
Secant line =
S/
t
D
I
S
P
L
A
C
E
M
E
N
T
time
S
t
t
1
t
2
s
1
s
2
Secant
line
D
I
S
P
L
A
C
E
M
E
N
T
time
s
1
t
1
Finding velocity
at
point t
1
, s
1
(instantaneous velocity)
Draw a tangent line at the point
t
S
Find the slope of
the tangent line
Instantaneous velocity
equals the slope of
the tangent line
DISPLACEMENT, VELOCITY &
CONSTANT ACCELERATION
•
The
velocity
of an object at an instant can be found by
determining the
slope of a tangent line
drawn at a point to
a graph of
displacement versus time
for the object.
•
If several instantaneous velocities are found and plotted
against time the graph of
velocity versus time is a straight
line
if the object is experiencing
constant acceleration
.
•
The slope of the straight line velocity versus time graph is
constant and since acceleration can be determined by the
slope of a velocity
–
time graph, the acceleration is
constant.
•
The graph
acceleration versus time
for a
constant
acceleration
system is a
horizontal line
. (A slope of zero
since constant acceleration means that acceleration is not
changing with time!)
D
I
S
P
L
A
C
E
M
E
N
T
Time
V
E
L
O
C
I
T
Y
Time
A
C
C
E
L
E
R
A
T
I
O
N
Time
S
t
t
v
Slope of a tangent drawn to a point on
a displacement vs time graph gives
the instantaneous velocity at that point
Slope of a tangent drawn to a point on
a velocity vs time graph gives the
instantaneous acceleration at that point
PLOT OF INSTANTANEOUS
VELOCITIES VS TIME
MEASURING VELOCITY & ACCELERATION
•
VELOCITY IS MEASURED AS DISPLACEMENT PER
TIME. UNIT FOR THE MEASUREMENT OF VELOCITY
DEPEND ON THE SYSTEM USED. IN THE MKS SYSTEM
(METERS, KILOGRAMS, SECONDS) IT IS DESCRIBED IN
METERS PER SECOND.
•
IN THE CGS SYSTEM (CENTIMETERS, GRAMS,
SECONDS

ALSO METRIC) IT IS MEASURED IN
CENTIMETERS PER SECOND.
•
IN THE ENGLISH SYSTEM IT IS MEASURED AS FEET
PER SECOND.
•
ACCELERATION IN THE MKS SYSTEM IS EXPRESSED
AS METERS PER SECOND PER SECOND OR METERS
PER SECOND SQUARED.
•
IN CGS UNITS IT IS CENTIMETERS PER SECOND PER
SECOND OR CENTIMETERS PER SECOND SQUARED. IN
THE ENGLISH SYSTEM FEET PER SECOND PER SECOND
OR FEET PER SECOND SQUARED ARE USED.
GRAVITY & CONSTANT ACCELERATION
•
Gravity is the most common constant acceleration
system on earth. As object fall under the influence of
gravity (free fall) they continually increase in velocity
until a terminal velocity is reached.
•
Terminal velocity refers to the limiting velocity
caused by air resistance. In an airless environment the
acceleration provided by gravity would allow a
falling object to increase in velocity without limit
until the object landed.
•
In most problems in basic physics air resistance is
ignored. In actuality, terminal velocity is related to air
density, surface area, the velocity of the object and
the aerodynamics of the object (the drag coefficient).
CLICK
HERE
19.6 m
44.1 m
78.4 m
19.6 m/s
2.0 sec
29.4 m/s
3.0 sec
39.2 m/s
4.0 sec
CALCULATING AVERAGE VELOCITY
•
Average velocity for an object moving with uniform
(constant) acceleration can be calculated in two ways.
•
(1) average velocity = the change in displacement
(displacement traveled, s) divided by the change in
time ( t). (s is the symbol used for displacement)
•
(2) average velocity = the sum of two velocities divided
by two (an arithematic average).
CALCULATING INSTANTANEOUS
VELOCITY
•
Instantaneous velocity can be found by taking the
slope of a tangent line at a point on a displacement
vs. time graph (as previously discussed).
•
Instantaneous velocity can also be determined from
an acceleration vs. time graph by determining the
area under the curve.
•
For constant acceleration systems, the acceleration
times the time (a x t) plus the original velocity (v
0
)
also gives the instantaneous velocity.
A
C
C
E
L
E
R
A
T
I
O
N
Time
t
1
AREA UNDER THE CURVE
(acceleration x time)
GIVES THE INSTANTANEOUS
VELOCITY AT TIME
t
1
CALCULATING DISPLACEMENT
•
Displacement of a body in constant
acceleration can be found in two ways.
•
Displacement is given by the area under a
velocity vs. time graph.
•
Displacement can also be found using the
follow equation where s
i
= instantaneous
displacement, v
o
= the original velocity of the
object, a = the constant acceleration and
t = elapsed time.
V
E
L
O
C
I
T
Y
Time
t
1
AREA UNDER THE CURVE
(velocity x time)
GIVES THE INSTANTANEOUS
DISPLACEMENT AT TIME
t
1
CALCULATING VELOCITY &
ACCELERATION FROM
DISPLACEMENT VS. TIME
•
The instantaneous velocity of an object can be found
from a displacement versus time graph by measuring
the slopes of tangent lines drawn to points on the
graph.
•
Since the derivate of an equation gives the formula for
calculating slopes, the derivative of the displacement
versus time equation will give the equation for velocity
versus time.
•
Additionally, the slope of an velocity versus time curve
is the acceleration. Therefore, the derivative of the
velocity versus time equation gives the acceleration
versus time relationship.
The first derivative of displacement versus time
gives the instantaneous velocity in terms of time.
The first derivative of velocity versus time gives
the instantaneous acceleration in terms of time.
CALCULATING VELOCITY &
DISPLACEMENT FROM ACCELERATION
•
The instantaneous velocity of an object can be
determined from the area under an acceleration
versus time graph.
•
Since the integration of an acceleration versus
time equation gives the area under the curve, it
also gives the velocity.
•
The area under a velocity versus time graph gives
the displacement. Therefore, the integral of the
velocity versus time equation gives the
displacement versus time equation.
The constant
is the original
velocity (V
0
)
The constant C is the original displacement of the object
If displacement is not measured from zero
Displacement
vs
time
Velocity
vs
time
slope
Acceleration
vs.
time
slope
Area
under
curve
Area
under
curve
Displacement
vs
time
derivative
Velocity
vs
time
Acceleration
vs.
time
derivative
integral
integral
ACCELERATED MOTION SUMMARY
•
V
AVERAGE
=
s/
琠㴠⡖
2
+ V
1
) / 2
•
V
INST.
= V
ORIGINAL
+
at
•
S
INST
= V
0
t +
½ at
2
•
Instantaneous velocity at a point equals the slope of a tangent
line drawn at that time point on a displacement vs. time graph
•
Instantaneous acceleration at a point equals the slope of a
tangent line drawn at that time point on a velocity vs. time
graph.
•
The derivative of a displacement vs. time equation gives the
instantaneous velocity.
•
The derivative of a velocity vs. time equation gives the
instantaneous acceleration.
•
The integral of an acceleration vs. time equation gives the
instantaneous velocity.
•
The integral of an velocity vs. time equation gives the
instantaneous displacement.
PUTTING EQUATIONS TOGETHER
•
Often problems involving uniformly accelerated
motion do not contain a time value. When this occurs
these problems can be solved by combining equations
which are already known.
•
To simplify the algebra, V
0
will assumed to be zero.
Therefore,
V
i
= V
O
+
at
becomes V
i
=
at
and
S
i
= V
0
t +
½ at
2
becomes S
i
=
½ at
2
.
•
Solving V
i
=
at
for t we get t =V
i
/a. Substituting into
S
i
=
½ at
2
gives S
i
=
½ a(
Vi/
a
)
2
or by simplifying
the equation S
i
=
½ (
Vi
2
/
a
)
•
If V
0
is not equal to zero the equation becomes
S
i
=
½ (
Vi
2
–
V
o
2
) /
a
(time is not required to solve
this equation!)
•
In the next program the equations and
relationships developed here will be used to solve
one dimensional, uniform acceleration problems.
•
Free fall problems will be included since they are
the most common examples of constant
acceleration .
•
Problem involving variable acceleration and use
to derivatives and integrals for their solution will
be covered.
•
Math concepts are required and if the program
on Math for Physics has not yet been viewed, it
may be a good idea to do so!
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