Question

Is proportionality Everything? A great many laws of physics and chemistry are expressible as proportionalities. Give at least one example of a function that occurs in the sciences that is not a proportionality.

Short answer

The equations of motion are not proportionalities $v=u+at$.

Step-by-step solution

Step 1. Apply the concept of variation.

Two quantities are directly proportional if on increasing one quantity the other quantity also increases. And on decreasing one quantity the other quantity also decreases.

Mathematically it is expressed as $y=kx$ where y and x are quantities and k is the constant of proportionality along with the condition $k\ne 0$.

It is read as y varies directly with x or y is directly proportional to x.

Two quantities are inversely proportional if on increasing one quantity the other quantity also decreases. And on decreasing one quantity the other quantity increases.

Mathematically it is expressed as $y=\frac{k}{x}$ where y and x are quantities and k is the constant of proportionality along with the condition $k\ne 0$.

It is read as y varies inversely with x or y is inversely proportional to x.

Step 2. Example of Inverse variation and direct variation.

To denote the statement that the distance covered by a car is directly proportional to the fuel consumption, denote fuel consumption by f and distance covered by d. Therefore, it is expressed as $f=kd$ where k is the constant of proportionality along with the condition $k\ne 0$.

To denote the statement that a number is inversely proportional to the square of other number, denote first number by pand second number by q. Therefore, it is expressed as $p=\frac{k}{q^2}$ where k is the constant of proportionality along with the condition $k\ne 0$.

Step 3. Equations that are not proportionalities.

If the position of an object varies with regard to a reference point, it is said to be in motion with respect to that point, but if it does not, it is said to be at rest with respect to that point.

There are certain basic equations related words distance, displacement, speed, velocity, and acceleration of the body by the equation called equations of motion.

The three conventional equations of motion, often known as the laws of constant acceleration, in the case of motion with uniform or constant acceleration (one with equal change in velocity in equal intervals of time).

These equations govern the motion of a particle by containing the values displacement(s), velocity (initial u and final v), time(t), and acceleration(a).

Only when a body's acceleration is constant and motion is on a straight line can these equations be used. The three equations are as follows:

$\begin{array}{c}v=u+at\\ v^2-u^2=2as\\ s=ut+\frac{1}{2}a{t}^2\end{array}$.

These equations does not contain any constant of proportionality.

Also, there is no direct and inverse relationship between the variables.

Step 4. Example of equation of motion.

We can forecast the distance travelled by a body if we know its initial and end velocities as well as its acceleration, even if we don't know how long it took to go.