2.2 Applications of Power Functions

Apollo 15

Introduction

In 1687 Isaac Newton published Principia, one of the most important books in the history of science.
A rare first edition copy of Principia sold at auction for $3.7 million in 2016. In Principia Newton formulated his three laws of motion, expanded Kepler's work on planetary orbits, laid the foundations for classical mechanics and, for the first time ever, gave a mathematical description of gravity.

Newton wrote that the gravitational force between two objects is "proportional to the product of their masses and inversely proportional to the square of the distance between them".

In this section we'll spend time trying to figure out what Newton meant by that and uncover a few other applications of power functions along the way.

The Standard Power Function Model

Most power functions applications require us to scale the basic form $f(x)=x^{p}$ by a factor of $k$. The resulting equation

[ f(x) = k x^{p} ]

is called the standard power function model.

To understand how $k$ impacts the model let's focus on the effect it has on a single point.

To use the interactive figure visit   https://www.geogebra.org/m/enm1qAJj

This causes a vertical scaling of the graph by a factor of $k$, so that any power functions of the form $f(x)=k x^{p}$ will pass through the point$(1, k)$.

In many cases the scaling factor $k$ is composed of more than one constant. For instance, the volume of a sphere is a function of the of the radius and is given by $V=\frac{4}{3} \pi r^{3}$. Here the constant $k$ includes both the fraction $\frac{4}{3}$ and $\pi$, that is $k=\frac{4}{3} \pi$.

Applications of Power Functions

Frequently we will find that the only difference between applications is the value of the scaling constant $k$ and, of course, the context. The next three examples illustrate this fact. All three are completely different applications, yet each one is modeled by $f(x)=kx^{1/2}$ for differing values of $a$.

The goal here is not to master the applications, but simply to become familiar with the terminology and maybe even begin to appreciate the usefulness of power functions. All you will be asked to do at this point is evaluate the functions.

Tugboats

Photo by James Abbot on Flickr

Light, fast boats are often designed to skim, or plane, across the surface of the water. However, when a large, heavy boat moves it must push the water out of its way. This creates a wave at the bow (front) of the boat. As it tries to go faster and faster the wave gets bigger and bigger.

When the length of this wave matches the length of the boat, then the boat has reached its hull speed $V$, which is generally considered to be the fastest it can go. The hull speed of a large ship doesn't depend at all on the size of its motor(s) or how heavy it is. Hull speed is only a function of the ship's length and is given by

[ V=1.34 L^{1/2} ]

where $V$ is the hull speed in knots and $L$ is the length of the boat at the water line in feet.

One surprising aspect of this model is that it not only works for large boats, but for waterfowl like ducks and geese since they make bow waves too.
Photo by Capri23auto from Pixabay If a duck has length of 1.25 feet at the water line, Then its predicted top speed swimming is

[ V=1.34(1.25)^{1/2}\approx 2 \text{knots} ]

Pendulums

A pendulum is a object that swings from a fixed point. A playground swing and a grandfather clock are two examples of pendulums. In one tower of the Oregon Convention Center, the 900 pound Principia pendulum hangs from a 70 foot long cable as it swings.

Surprisingly, the amount of time that it takes a pendulum to swing back and forth once (called its period) doesn't depend at all on how heavy it is. The period of a pendulum depends only on the length of the cable and is given by

[ P=1.11 \thinspace L^{1/2} ]

where $P$ is the period in seconds and $L$ is the length of the pendulum in feet.

⊕ Photo by Lisa Brideau on Flickr

Pendulums in grandfather clocks are often 39 inches long. Since 39 inches is 3.25 feet, this gives them a period of

[ P=1.11\thinspace (3.25)^{1/2}\approx 2.0024 \text{ seconds} ]

If the entire period is 2 seconds, then each swing to the left or to the right lasts almost exactly 1 second, just what a clockmaker would want.

Falling Bodies

Photo by Monika Neumann on Pixabay

Italian scientist Galileo Galilei is reported to have dropped two spheres of different masses from the top of the Leaning Tower of Pisa and observed that both hit the ground at the same time. He concluded that the time it takes an object to fall is independent of its mass. In other words, if air resistance is removed, a light object, like a feather, will fall just as fast as a heavy object, like a hammer.

On Earth, the time it takes an object to fall, ignoring air resistance, is given by

[ t=0.25 d^{1/2} ]

where $t$ is time in seconds and $d$ is the distance in feet. This was famously verified on the Moon by the astronauts of Apollo 15 in 1971, who captured the event on video.

In 1930, Chicago Cubs' baseball player Gabby Hartnett caught a baseball that had been dropped 800 feet out of a blimp before a pre-season game in Los Angeles. Ignoring air resistance, the ball would have fallen for [ t=0.25(800)^{1/2}\approx 7 \text{ seconds} ]

According to reports, the blimp tossed out another ball which he caught as well. The baseball reached a top speed of about 95 miles per hour.

Direct Variation

Many applications, especially in science and engineering, are expressed in the language of variation. Our focus here is on the vocabulary and converting between verbal and symbolic descriptions. Solving equations is left for the next section.

Two quantities $x$ and $y$ are said to vary directly Variation is sometimes called proportionality. The phrase "$y$ varies directly with $x$" means the same as "$y$ is directly proportional to $x$". if $y$ is a constant multiple of $x$. Algebraically this means that

[ y = k \thinspace x ]

for any constant $k$.

In a variation problem the number $k$ is called the constant of variation. Whenever $x$ increases by $1$ then $y$ will increase by a factor of $k$.

While we normally think of $x$ and $y$ as single variables, they could also be algebraic expressions. For instance, Kepler's third law describes a directly proportional relationship between a square and a cube. It states that the square of the time it takes a planet to orbit the Sun (called its period) varies directly with the cube of its distance from the Sun. The equation is $P^2 = k \thinspace D^{3}$.

In this case the two quantities that are proportional are $P^2$ and $D^{3}$.

Of particular interest to us are equations of the form $y = k \thinspace x^{p}$, where $y$ varies directly with the $p$-th power of $x$. Note that this is nothing more than our standard power function model with a different name.

Recall, for instance, that the volume of a sphere is given by the power function $V=\frac{4}{3} \pi r^{3}$ where $r$ is the radius. Using the terminology of variation, we can also say that volume is proportional to the cube of the radius, with a constant of variation of $\frac{4}{3} \pi$.

Inverse Variation

When $y = k x^{p}$ and the power is negative, there is another phrase that is used to describe the relationship between $x$ and $y$. If

[ y = k \thinspace x^{-n} = k \frac{1}{x^{n}} , n>0 ]

then $y$ varies inversely with $x^n$. Notice that being inversely proportional to $x^n$ is the same as being directly proportional to $x^{-n}=\frac{1}{x^{n}}$. For example, if $y = 4 x^{-3}$, then $y$ is inversely proportional to $x^{3}$.

If two quantities vary inversely, then one increases when the other decreases, and vice versa. This is in contrast to direct variation where both quantities increase, or decrease, at the same time.

Two quick examples should illustrate the difference. When cooking rice, the amount of water needed is directly proportional to the amount of rice. If you want to cook more rice, you'll need more water. However, the time it takes to drive a car around a race track is inversely proportional to the speed of the car. The faster the car travels, the less time it will take to do a lap.

Boyle's Law

One example of inverse variation is Boyle's Law. According to Boyle's Law, if the temperature of a gas in a sealed container is held constant, then the volume of the gas is inversely proportional to the pressure applied. Algebraically this means that

[ P=k \thinspace V^{-1} ]

where $P$ is pressure, $V$ is volume, and the constant $k$ depends on the initial condition of the gas.

To see why this might be useful, suppose that a medical syringe contains $0.1$ cubic centimeter of air when it is closed and that $k=0.103$ kilogram-centimeters. Then the pressure in the syringe is

[ P=0.103(0.1)^{-1}=1.03 \text{ kilograms per square centimeter} ]

If the plunger of the syringe is pulled back, increasing the volume to $5.0$ cubic centimeters, then the pressure drops to

[ P=0.103(5.0)^{-1}=0.0206 \text{ kilograms per square centimeter} ]

It is this drop in pressure that causes the suction needed to draw a sample of blood.

Combining Direct and Inverse Variation

Direct and inverse variation are frequently combined together. Here we will examine a few examples and translate the descriptions into equations.

Stopping Distance

Photo by Hannes215 on Pixabay

Physics shows that the stopping distance $d$ of a car is directly proportional to the square of its speed $v$ and inversely proportional to the friction $\mu$ between the tires and the road. If $v$ is measured in miles per hour and $d$ in feet, then

[ d = \frac{0.0336 v^2 }{\mu} ]

It is interesting to note that the size and weight of the vehicle are not part of the equation. More importantly, from a safety standpoint, is the fact that doubling the speed $v$ results in a quadrupling of the stopping distance $d$.

Suppose you are driving $60$ mph down a snowy road and suddenly slam on the brakes. When will you come to a complete stop? We now know the equation, but before we can use it we need the friction coefficient $\mu$. Finding the exact value of the friction coefficient is difficult, but traffic engineers have calculated approximate values for different surfaces.

In this case we will use $\mu = 0.2$, since the car is on snow. Substituting these values into our equation gives a stopping distance of

[ d = \frac{0.0336(60)^{2}}{0.2} \approx 605 \text{ feet} ]

So if you were traveling at 60mph down a snow covered road, it would take about 605 feet for your car to stop once you had applied the brakes.

Newton's Universal Law of Gravity

As a final example let's now turn our attention to Newton's law of gravity. Isaac Newton discovered that the force of gravity $F$ between two objects, like the Sun and the Earth, is proportional to the product of their two masses, $m_1$ and $m_2$, and inversely proportional to the square of the distance $d$ between them.

In equation format, we have

[ F = k \frac{m_1 m_2 }{d^{2}} ]

As a random fact, the units of $F$ are called "Newtons", in his honor. One Newton is roughly equal to the weight of an apple, about 1/4 of a pound.

Finding the Constant of Variation

Occasionally, it will be necessary to find the constant of variation. Whether dealing with direct or inverse variation, the process is basically the same: insert two known values into the equation and solve for $k$.

For instance, if $y$ varies directly with $x$ and $y=72$ when $x=12$, then we can find $k$ by solving $72=k\cdot12$ by dividing both sides by $12$ to get $k=6$. With this in hand, other values of $y$ can be calculated using the direct variation equation $y=6x$.

As another example, suppose $y$ varies inversely with $x$ and that $y=3$ when $x=14$. Then $3=\frac{k}{14}$ and, after multiplying by $14$, we find that $k=42$. So the inverse relationship between $x$ and $y$ is $y=\frac{42}{x}$.

Looking Ahead

In this section we were given equations that represented real life scenarios and were able to evaluate them. In the next section we will solve these equations for different parameters, and in the section after that we will create our own models.