The International System of Units
Quantity
Unit Name
Symbol
Length
meter
m
• Many SI Derived Units:
Time
second
s
1 Newton = 1 N = 1 kg · m/s2
Mass
kilogram
kg
• Three Basic Units (SI)
1 Watt = 1 W = 1 N · m = 1 kg · m2/s2
• Prefixes for SI Units:
1 kg = 1 × 103 grams
1 ps = 1 ×
10-12
Conversion Factor
seconds
• Changing Units:
1 min
=1
1 min = 60 s →
60 s
60 s
=1
1 min
⎛ 60 s ⎞
2 min = ( 2 min)(1) = ( 2 min)⎜
⎟ = 120 s
⎝ 1 min ⎠
R. Field 8/27/2013
University of Florida
PHY 2053
Factor
Prefix
Symbol
1012
tera-
T
109
giga-
G
106
mega-
M
103
kilo-
k
10-2
centi-
c
10-3
milli-
m
10-6
micro-
μ
10-9
nano-
n
10-12
pico-
p
Page 1
distance: meter
1791: one ten-millionth of a quadrant of the Earth…
--1889: platinum-iridium bar
--1983: The meter is the length of the path traveled by
light in vacuum during a time interval of
1 / (299,792,458) of a second
(i.e., by definition, c = 299,792,458 m/s)
R. Field 8/27/2013
University of Florida
PHY 2053
Page 2
time: second
Since long ago (Egyptians and Greeks): 1 day = 24 hours
1 s = (1/24) x (1/60) x (1/60) of the full Earth turn
However,
the Earth
rotation
period
varies by
a few ms…
1967: The second is the duration of 9,192,631,770
periods of the radiation emitted by caesium-133 atom
R. Field 8/27/2013
University of Florida
PHY 2053
Page 3
mass: kilogram
1799: 10x10x10 cm3 of water
1 u = 1.66053886 × 10
--1899: platinum-iridium cylinder (d=h=39 mm)
-27
kg
atomic mass unit (u):
• 12 u = mass of 12C atom
• universal, reproducible,
nor requiring an artifact
Density: The density of a
material, ρ, is the mass per
unit volume:
ρ = m/V
ρ(water) = 1.00 g/cm3
R. Field 8/27/2013
University of Florida
PHY 2053
Page 4
1-d Motion: Position & Displacement
• The x-axis:
We locate objects by specifying their
position along an axis (in this case x-axis).
The positive direction of an axis is in the
direction of increasing numbers. The
opposite is the negative direction.
• Displacement:
x(t)
The change from position x1 to position x2
is called the displacement, Δx.
Δx = x2 –x1
The displacement has both a magnitude,
|Δx|, and a direction (positive or negative).
time t
• Graphical Technique:
A convenient way to describe the motion
of an object is to plot the position x as a
function of time t (i.e. x(t)).
R. Field 8/27/2013
University of Florida
PHY 2053
Page 5
1-d Motion: Average Velocity
• Average Velocity
v = vave
The average velocity is defined to be
the displacement, Δx, that occurred
during a particular interval of time, Δt
(i.e. vave = Δx/Δt).
Δx x(t 2 ) − x(t1 ) x2 − x1
=
=
=
t 2 − t1
t 2 − t1
Δt
• Average Speed
The average speed is defined to be the
magnitude of total distance covered
during a particular interval of time, Δt
(i.e. save = (total distance)/Δt).
R. Field 8/27/2013
University of Florida
PHY 2053
Page 6
Average Velocity: Example Problem
Final Position
3:56 pm
Initial Position
3:14 pm
xf = -26 km
x=0
W
xi = 3 km
x-axis
xE = 10 km
E
A train that is initially at the point xi = 3 km at 3:14 pm travels 7 km to the East
to the point xE = 10 km. It then reverses direction and travels 36 km to the
West to the final point xf = -26 km arriving at 3:56 pm. What is the train’s
average velocity (in km/h) for this trip?
v = vave
Δx x(t f ) − x(ti ) x f − xi
− 26km − (3km)
− 29km
=
=
=
=
=
≈ −41.43km / h
Δt
t f − ti
t f − ti
42 min× (1h / 60 min)
0. 7 h
Note that the displacement Δx is equal to the average velocity times Δt.
Δx = vave Δt = (−41.43km / h)(0.7h) ≈ −29km
R. Field 8/27/2013
University of Florida
PHY 2053
Page 7
Average Speed: Example Problem
Final Position
3:56 pm
Initial Position
3:14 pm
xf = -26 km
x=0
W
xi = 3 km
x-axis
xE = 10 km
E
A train that is initially at the point xi = 3 km at 3:14 pm travels 7 km to the East
to the point xE = 10 km. It then reverses direction and travels 36 km to the
West to the final point xf = -26 km arriving at 3:56 pm. What is the train’s
average speed (in km/h) for this trip?
save =
d
7 km + 36km
43km
=
=
≈ 61.43km / h
Δt 42 min× (1h / 60 min) 0.7h
Note that the total distance d is equal to the average speed times Δt.
d = save Δt = (61.43km / h)(0.7 h) ≈ 43km
R. Field 8/27/2013
University of Florida
PHY 2053
Page 8
1-d Motion: Instantaneous Velocity
x
x(t+Δt)
x(t)
R. Field 8/27/2013
University of Florida
{
shrink Δt
t
t+Δt
t
Δt
PHY 2053
Page 9
1-d Motion: Instantaneous Velocity
x
x(t+Δt)
x(t)
R. Field 8/27/2013
University of Florida
{
shrink Δt
t+Δt
t
t
Δt
PHY 2053
Page 10
1-d Motion: Instantaneous Velocity
x
x(t+Δt)
x(t)
R. Field 8/27/2013
University of Florida
t+Δt
{
shrink Δt
t
t
Δt
PHY 2053
Page 11
1-d Motion: Instantaneous Velocity
x
x(t) x(t+Δt)
R. Field 8/27/2013
University of Florida
{
shrink Δt
t t+Δt
Δt
t
PHY 2053
Page 12
1-d Motion: Instantaneous Velocity
x
tangent line at t
x(t)
Δx dx
=
v(t ) = lim
dt
Δt →0 Δt
R. Field 8/27/2013
University of Florida
Instantaneous velocity v(t) is
slope of x-t tangent line at t
t
t
The velocity v(t) is the rate of change
of x(t) with respect to t at time t.
PHY 2053
Page 13
1-d Motion: Acceleration
• Acceleration
a = aave
When a particles velocity changes, the
particle is said to undergo acceleration
(i.e. accelerate).
v
Δv v(t 2 ) − v(t1 ) v2 − v1
=
=
=
Δt
t 2 − t1
t 2 − t1
v(t)
v2
• Average Acceleration
Hello
Δv “rise”
v1
The average acceleration is defined to
be the change in velocity, Δv, that
occurred during a particular interval of
time, Δt (i.e. aave = Δv/Δt).
• Instantaneous Acceleration
a
Δv
v
The acceleration a(t) is the rate of
change of v(t) with respect to t at time t.
v(t)
Δv dv
=
a (t ) = lim
Δt →0 Δt
dt
R. Field 8/27/2013
University of Florida
Instantaneous acceleration a(t) is
slope of v-t tangent line at t
PHY 2053
Page 14
1-d Motion: Summary
• Instantaneous Velocity
The velocity v(t) is the rate of change
of x(t) with respect to t at time t.
Δx dx
v(t ) = lim
=
dt
Δt →0 Δt
a (t ) =
• Instantaneous Acceleration
The acceleration a(t) is the rate of
change of v(t) with respect to t at time t.
Δv dv
=
a (t ) = lim
Δt →0 Δt
dt
R. Field 8/27/2013
University of Florida
dv d dx d 2 x
=
=
dt dt dt dt 2
v
v(t)
Instantaneous acceleration a(t) is
slope of v-t tangent line at t
PHY 2053
Page 15