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Non-Projectile Quadradic Examples
Name: Denise H.
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: N/A
Question:
I am looking for some real life applications of solving
quadratic equations by finding square roots. Other than the typical
falling object model.
Replies:
Denise,
There are a variety.
One that comes to mind is gravitational force at planetary and stellar
distances. Newton's force formula at a long distance is an inverse-square
law. If you know the masses of a planet and its moon, as well as the
force between them, you can calculate the distance with inverting a
fraction and then taking the square root: (Force)=G(mass 1)(mass
2)/(distance)^2, where
G=6.67x10^-11 J.m^2/kg^2.
A similar problem deals with centripetal force and velocity. For any
object moving along a circular path, the force provided by the outside
world must be just large enough to equal
(Net Force)=(mass)(speed)^2/(path radius). If you know the mass of the
Moon, the radius of its orbit, and the pull the Moon feels from the Earth,
you can use the centripetal force equation to
solve for the speed of the Moon. This same relation applies to a rock on
a string spinning in a circle. Tension in the string is the force.
Another relation with a squared term is almost any circular or square area
problem. A good example is the force exerted on such a surface by a
liquid or gas. The pressure of the fluid multiplied by the area of the
surface is the force exerted on that surface. The pressure of the air is
14.5 pounds per square inch, or 10^5 Newtons per square meter. A circular
surface feels 30 pounds of pressure from the air. What is the radius of
this circle?
Kinetic energy, or energy of motion, is another option. The formula for
kinetic energy is KE=(1/2)(mass)(speed)^2. Any distance-squared,
speed-squared, or 1/(distance-squared) relation will do just fine.
Dr. Ken Mellendorf
Physics Instructor
llinois Central College
There are two questions here: 1. "solving quadratic equations by finding
square roots" and
2. Applications "other than the typical falling object model."
Regarding 1. There are many ways to solve quadratic equations (equations in
which the highest power of the unknown or unknowns is 2) other than by
directly taking square roots. The familiar "quadratic formula", "completing
the square", so called "Newton's method", graphing, and a variety of
numerical methods. Most quadratic equations are NOT in the form of a
"perfect square" so that taking the square root directly provides the
solution.
Regarding 2. Applications of quadratic equations are so numerous in all of
science and mathematics that it is difficult to know where to start a list.
Any problem involving circles, ellipses, and their 3 dimensional analogs
the sphere and ellipsoid (of which there must be millions) all involve
quadratic equations. The kinetic energy is defined as K.E.= 1/2 m*v^2 where
'v' is the velocity, so any problem involving kinetic energy (of which there
must be billions) involve quadratic equations. Quadratics appear in all
sorts of optical and acoustic phenomena (the parabolic mirror, or parabolic
microphone), the rates of many chemical reactions vary as the square of the
concentration of one or more of the reagents, a Gaussian distribution is of
the form:
~ exp(-x^2) which is quadratic, so any application of statistics will
involve quadratic equations in one form or another.
Vince Calder
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Update: June 2012
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