Download Mathematical Reasoning

Mathematical Reasoning / Problem Solving Skills
1.
Recognizing proportionalities inside equations
A.
Direct proportionalities; linear functions
Suppose A = k
.
B, where A, B are variables and k is an empirical constant
This is equivalent to writing A α B
.
ex.
Hubble’s Law: Velocity = H
B.
Direct proportionalities; non-linear functions
Suppose A = k
.
Distance or V α D
Bn, where n = 2, 3, 4, …
This is equivalent to writing A α Bn
ex.
Stefan-Boltzman Law: Flux = σ T4 or Flux α T4
C.
Inverse proportionalities; linear functions
Suppose A = k / B
This is equivalent to writing A α 1 / B
ex.
Wien’s Law: Temperature = k / λ max. or Temperature α 1 / λ max.
D.
Inverse proportionalities; non-linear functions
Suppose A = k / B2
This is equivalent to writing A α 1 / B2
II.
ex.
Newton’s Law of Gravity: F = G M m/ r2 or F α 1 / r2
ex.
Inverse-square Law: Brightness α 1 / distance2
Ratios and Proportions: To solve comparative problems for two related objects or conditions
Suppose two stars are identical, but star B is 4 times farther away than star A. What is the brightness
ratio between star A and star B?
Brightness A α 1 / (distance A)2
Brightness B α 1 / (distance B)2
Brightness A = 1 / (distance A)2 = (distance B)2 = (4 distance A)2 = 42 = 16
Brightness B
1 / (distance B)2
(distance A)2
(distance A)2
III.
Dimensional Analysis: In performing calculations, we must make sure that all units agree
(and appropriately cancel out).
Values (i.e., numbers) have units (i.e., dimensions) attached!
ex.
Suppose we want to calculate how far light travels in one year’s time (one light year):
distance = rate
.
time
= 186,000 mi/sec
.
60 sec/min
.
60 min/hr
.
24 h/day
.
365.25 day = 6 . 1012 mi [6 trillion mi]
A common source of error – to use a formula (e.g., Newton’s Law of Gravity or Kepler’s Third Law)
where you are to input a distance (in m), but the value you substitute is in km.
By dimensional analysis, you must first convert km to m: (1 km = 103 m)
***************************************
Units:
scientists use MKS system of units:
Distances = meters (or prefix thereof):
10-9 = nm
10-6 = µm
10-3 = mm
103 = km
mass = kilograms
time = seconds
106 = M [Hz; yr]
109 = G [Hz; yr]