When you
study science or geometry, you come across several formulas in the form of a =
kx and b = ½kx2. For example,
the velocity of an object starting at rest with a constant velocity can be
determined by multiplying acceleration and time, or v = at, and the distance of
the same object can be found by taking half of its acceleration multiplied by
the time squared, or d = ½at2.
For another example, the momentum of an object can be determined by multiplying
its mass times its velocity, or p = mv, and the kinetic energy of an object can
be found by taking half of its mass times its velocity squared, or E = ½mv2.
Sometimes
the equations need to be manipulated to be in the form of a = kx and b = ½kx2. For example, the circumference of a circle is
C = 2πr and the area of a circle is A = πr2, but if we substitute τ
= 2π, then C = τr and A = ½τr2.
For another example, the area of a triangle is A = ½bh, but if height is
defined as a ratio of the base, then h = kb and A = ½kb2.
Other
equations in the form of a = kx and b = ½kx2 include angular
velocity and displacement, angular momentum and rotational energy, force and
potential energy of a spring, electric flux and energy density, and electric
charge and electric energy:
a
= kx
|
b
= ½kx2
|
Velocity
v = at
|
Distance
d = ½at2
|
Angular Velocity
ω = αt
|
Angular displacement
θ = ½αt2
|
Momentum
p = mv
|
Kinetic Energy
E = ½mv2
|
Angular Momentum
L = Iω
|
Rotational Energy
K = ½Iω2
|
Force of a Spring
F = kx
|
Potential Energy of a Spring
E = ½kx2
|
Electric Flux Density
D = εE
|
Electric Energy Density
Q = ½εE2
|
Electric Charge
Q = CV
|
Electric Charge Energy
E = ½CV2
|
Circumference of a Circle
C = 2πr = τr
|
Area of a Circle
A = πr2 = ½τr2
|
Height of a Triangle
h = kb
|
Area of a Triangle
A = ½bh = ½kb2
|
In fact, if
we lift the requirement that k must be a constant, we can include even more
formulas to the form of b = ½kx2.
Another pair of similar yet unrelated formulas is gravitational force (F
= Gm1m2/r^2) and electric force (F = kq1q2/d^2). Gravitational force can be rearranged to ½Gm1m2
= b = ½Fr2 and electric force can be rearranged to ½kq1q2
= b = ½Fd2. Even Einstein’s
famous theory of relativity E = mc2 can be manipulated to ½E = ½mc2.
Those
familiar with calculus may recognize that the derivative of b = ½kx2
is a = kx, which means a is a rate of b with respect to x. For example, since d = ½at2 and v
= at, velocity is a rate of distance with respect to time. A good way to visualize this is to use the
two geometry formulas of the circle and triangle. Since A = ½τr2 and C = τr for a
circle, the circumference is a rate of its area with respect to its radius,
which means that the area of a circle is the sum of all the circumferences it
contains.
Area of a Circle = Sum of the Circumferences
Since A =
½kb2 and h = kb for a triangle, the height is a rate of its area
with respect to its base, which means that the area of a triangle is the sum of
all the heights it contains.
Area of a Triangle = Sum of the Heights
Although calculus
can be used to explain the relationship between b = ½kx2 and a = kx,
it cannot explain the high frequency of which these formulas appear in
nature. If there were some relationship
between all these formulas we might be tempted to attribute the high frequency
to a coincidence, but these equations appear in completely unrelated and
different fields of math and science, from velocity to circles, from springs to
electric fields, and from momentum to triangles. So many different formulas in the form of b =
½kx2 and a = kx cannot be a result of random chance, but rather must
be the result of an Intelligent Designer of an orderly universe.
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