# Proportionality to powers

**Proportionality to powers**

The direction proportionality to powers is when, one variable may be directly proportional to a power of the other variable. An example of this rule comes from kinetic energy where the object is proportional to the square of its speed.

Therefore if the speed of an object ** doubles** its kinetic energy will be

**greater (2² = 4) etc.**

*four times***Equations and square proportion**

If a quantity ** y** is directly proportional to the square of another quantity

**this is linked by writing the following expression:**

*x,**y ∝ x²*

This expression can then be converted to the following equation:

*y = kx²*

In this equation, *k* is the **constant of proportionality**.

Rearranging the equation will then mean that

*k* = .

Therefore the ratio between *y* and x² is constant. However, keep in mind that *x* and *y* are variables.

Example 1

An example of *b* being directly proportional to *a*² therefore:

*b = ka²*

if *a =* 2 and *b = 16*

Note this equation can also be written as

**Inverse proportionality to powers**

The opposite to proportionality can also be said in that a variable can be inversely proportional to a power of the other variable.

For example, the electrical resistance *R* of a metre of wire is inversely proportional to the square of its diameter *d*.

This can be written into the following expression:

This expression can then be converted to the following equation:

Example 2:

If thee electrical resistance of a metre of wire with a diameter of 2 mm is 1.2 Ohms what is *k?*

Insert the variable to the following equation: