Quadratic Proportionality
Direct quadratic proportionality describes a relationship in which a quantity \( y \) is proportional to the square of another quantity \( x \). This is expressed mathematically as: $$ y = k \cdot x^2 $$ where \( k \) is a constant. In this relationship, the ratio of \( y \) to \( x^2 \) remains constant: $$ \frac{y}{x^2} = k $$ On a graph, this relationship forms a parabola with its vertex at the origin.
Consider a simple example: an object falling freely under the force of gravity.
The formula for uniformly accelerated motion serves as a real-world example of direct quadratic proportionality.
$$ s = \frac{1}{2} g t^2 $$
Here, \( s \) represents the distance traveled (in meters, m), \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \) on Earth), and \( t \) is the elapsed time (in seconds, s).
According to this formula, the distance traveled (\( s \)) does not increase linearly with time (\( t \)) but instead follows a quadratic pattern.
In other words, the distance traveled is proportional to the square of the time elapsed.
In the absence of friction or air resistance, the distance traveled by a freely falling object is directly proportional to the square of the time.
A Practical Example
Imagine an object falling from a certain height.
We can calculate the distance it covers over time, assuming \( g = 9.8 \, \text{m/s}^2 \). Using the formula above, let’s determine \( s \) for different values of \( t \).
| Time (\( t \)) in seconds | Distance (\( s \)) in meters |
|---|---|
| 1 | \( 4.9 \) |
| 2 | \( 19.6 \) |
| 3 | \( 44.1 \) |
| 4 | \( 78.4 \) |
For example, after \( t = 1 \) second, the object falls \( s = \frac{1}{2} \cdot 9.8 \cdot 1^2 = 4.9 \) meters. After \( t = 2 \) seconds, the distance increases to \( s = \frac{1}{2} \cdot 9.8 \cdot 2^2 = 19.6 \) meters, and so on.
As shown, the distance increases quadratically with time.
If we calculate the ratio of distance \( s \) to the square of time \( t^2 \), it remains constant, verifying the direct quadratic proportionality: \( \frac{s}{t^2} \).
| Time (\( t \)) | \( s \) in meters | \( t^2 \) in seconds\(^2\) | \( \frac{s}{t^2} \) (\( \text{m/s}^2 \)) |
|---|---|---|---|
| 1 | 4.9 | 1 | 4.9 |
| 2 | 19.6 | 4 | 4.9 |
| 3 | 44.1 | 9 | 4.9 |
| 4 | 78.4 | 16 | 4.9 |
As we can see, the ratio \( \frac{s}{t^2} = 4.9 \, \text{m/s}^2 \) corresponds to half of the gravitational acceleration (\( \frac{g}{2} \)), as predicted by the formula.
Now, let’s plot a graph with \( t \) on the horizontal axis (time) and \( s \) on the vertical axis (distance).
The graph of quadratic proportionality forms a parabola with its vertex at the origin. This visually demonstrates the quadratic increase in distance over time.
Inverse Quadratic Proportionality
Another type of quadratic proportionality is inverse quadratic proportionality.
A quantity \( y \) is inversely proportional to the square of \( x \) if the product of \( y \) and \( x^2 \) remains constant. This can be expressed mathematically as: $$ y \cdot x^2 = k \quad \text{or} \quad y = \frac{k}{x^2} $$
Graphing the equation \( y = k / x^2 \) produces a hyperbolic curve that approaches but never touches the axes.
A well-known example of inverse quadratic proportionality in physics is the law of light intensity.
According to this law, the intensity of light \( I \) is inversely proportional to the square of the distance \( d \) from the source.
$$ I = \frac{k}{d^2} $$
Here, \( I \) represents light intensity (e.g., in lumens/m\(^2\)), \( d \) is the distance from the light source (in meters), and \( k \) is a constant based on the source’s power.
This relationship shows that light intensity diminishes rapidly with increasing distance, following an inverse quadratic law.
A Practical Example
Let’s consider a light bulb emitting a maximum intensity of \( k = 100 \) lumens/m\(^2\) at a distance of 1 meter (\( d = 1 \, \text{m} \)).
We can calculate the light intensity at different distances from the source.
| Distance (\( d \)) in meters | Light Intensity (\( I \)) in lumens/m\(^2\) |
|---|---|
| 1 | \( 100 \) |
| 2 | \( 25 \) |
| 3 | \( 11.11 \) |
| 4 | \( 6.25 \) |
For example, at a distance of 1 meter (\( d = 1 \, \text{m} \)), the intensity is \( \frac{100}{1^2} = 100 \) lumens/m\(^2\). At 2 meters (\( d = 2 \, \text{m} \)), it drops to \( \frac{100}{2^2} = 25 \) lumens/m\(^2\), and so on.
Notice that doubling the distance (\( d = 2 \)) reduces the intensity to a quarter (\( 25 \, \text{lumens/m}^2 \)), while tripling it (\( d = 3 \)) reduces the intensity to a ninth (\( 11.11 \, \text{lum ens/m}^2 \)).
If we plot this relationship, with \( d \) on the horizontal axis and \( I \) on the vertical axis, the resulting hyperbolic curve approaches the axes but never touches them.
As you can see, inverse quadratic proportionality provides a precise description of how intensity diminishes rapidly with increasing distance.
In conclusion, these mathematical relationships are not merely abstract formulas—they reflect the behavior of the physical world.
Whether describing free-falling objects or light intensity, graphs like parabolas and hyperbolas reveal the hidden patterns in nature’s laws.
