Transportation
Energy
Use
and
Cars
2
and
3:
Constant
Speed
Cruising
and
Rolling
Resistance
Take‐Home
Experiment
Write‐Up
Introduction
The
force
Fcar
required
to
keep
a
vehicle
moving
at
a
constant
speed
v
is
equal
to
the
combined
drag
force
at
the
speed.
1
Fcar = f r + D = µr mg + CD ρAv 2 2
(1)
Where
µr
is
the
coefficient
of
rolling
friction,
m
is
the
mass
of
the
vehicle
(and
occupant(s)),
g
is
acceleration
due
to
gravity,
CD
is
the
coefficient
of
drag,
ρ
is
the
density
of
air
displaced,
A
is
the
cross
sectional
area
of
the
cylinder
of
air
the
vehicle
displaces
and
v
is
the
constant
speed.
You
can
measure
€
Fcar
as
a
function
of
v
by
measuring
the
combined
drag.
This
can
be
done
simply
by
taking
your
foot
off
the
gas,
or
by
stopping
pedalling,
and
noting
the
rate
at
which
the
speed
is
reduced.
This
acceleration
a
can
be
multiplied
by
the
vehicle’s
mass
(don’t
forget
the
occupants!)
and
velocity
at
that
instant
to
give
the
mechanical
power
Pcar.
(2)
Note:
I
am
talking
magnitudes
only
here,
which
is
why
there
are
no
minus
signs
anywhere.
We
may
safely
assume
that
Pcar
is
positive.
The
following
equation
can
then
be
used
to
analyze
the
velocity
dependence
and
extract
the
coefficient
of
drag
and
rolling
friction.
(3)
Method
1. Measure
the
speed
as
a
function
of
time
for
your
vehicle.
For
a
car,
this
is
straightforward
if
there
are
two
of
you.
It
is
unsafe
to
attempt
this
alone!
For
bicycles
suggestions
include
buying
a
speedometer,
timing
along
a
marked
path,
or
using
a
small
digital
camera
that
shoots
video.
2. Pick
a
flat,
quiet
road
on
a
calm
day.
3. Accelerate
to
as
high
a
speed
as
is
safe
and
coast
to
a
standstill.
4. Using
a
spreadsheet
to
calculate
acceleration
and
force
from
velocity
and
time.
5. Calculate
the
power
as
a
function
of
velocity.
(Eq.
2)
6. Plot
speed
vs.
time
and
power
vs.
velocity.
7. Using
a
function
(Eq.
3)
and
by
manually
inputting
values
until
you
get
a
fit,
extract
the
coefficients
of
rolling
resistance
(µr)
and
drag
(CD).
SAFETY
NOTE:
•
•
It
is
not
necessary
to
drive
more
than
50
km/h
to
get
good
results.
Do
not
try
this
when
other
vehicles
are
around.
Physics
and
Astronomy
Outreach
Program
at
the
University
of
British
Columbia
•
Any
experimental
procedure
that
is
dangerous
will
receive
a
mark
of
zero.
Some
things
to
make
note
of:
•
•
•
•
Type
of
vehicle,
mass
(usually
given
inside
driver’s
door;
use
scales
for
you
and
bicycle)
and
frontal
area.
Date,
time
and
place
where
data
was
taken.
Note
local
wind
direction
and
estimate
speed
–
drop
a
piece
of
tissue
paper.
If
the
car
is
an
automatic,
coast
in
neutral.
Take
data
several
times
in
both
directions
to
account
for
any
slight
slope
or
wind.
Sample
Results:
A
bicycle
The
graph
below
shows
the
motive
force
required
to
maintain
a
bicycle
at
a
constant
velocity.
Data
were
taken
in
two
directions
(east
and
west)
and
the
coefficients
of
the
fitting
curve
(fit
to
Equation
1)
were
extracted
and
found
to
be:
CD
~
1
and
µR
~
0.0012.
Force
(N)
Velocity
(m/s)
Physics
and
Astronomy
Outreach
Program
at
the
University
of
British
Columbia
Sample
Results:
Comparison
of
two
cars
Data
below
was
collected
with
two
different
vehicles
(an
SUV
and
a
compact
car).
The
drag
coefficient
CD
~
0.2
and
the
coefficient
of
rolling
resistance
is
µR
~
0.1.
This
difference
in
the
curves
is
accounted
for
by
the
mass
of
the
vehicles.
Summary
The
coefficient
of
drag
and
of
rolling
resistance
for
a
car
or
a
bicycle
can
be
found
by
plotting
power
vs.
velocity.
The
coefficients
of
drag
and
rolling
resistance
for
most
vehicles
are
similar
but
the
power
required
to
maintain
a
constant
speed
vary
largely
because
of
the
mass
of
the
vehicles.
Physics
and
Astronomy
Outreach
Program
at
the
University
of
British
Columbia
References
Waltham,
C.E.,
&
Copeland,
B.
(1999).
Power
requirements
for
rollerblading
and
bicycling.
The
Physics
Teacher,
37(6),
379‐382.
Rachel
Moll
and
Chris
Waltham
(2009‐12‐08)
Physics
and
Astronomy
Outreach
Program
at
the
University
of
British
Columbia