# 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

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/

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!