Equations and Formulae

As someone who did science 'A' levels, I find the standard of science questions in the average pub quiz very disappointing. And to be fair, there is a limit to the amount of science that the average quiz contestant (who didn't do science 'A' levels) can be expected to know.

Every so often, you will be asked how something is calculated – or actually to calculate something. For example:

"In an electrical circuit, what quantity is given by the current multiplied by the voltage?"

Or:

"In an electrical circuit, if a current of three amperes flows across a potential difference of four volts, what is the amount of power generated (in watts)?"

Such questions are rather difficult to fit into the format of this website; but after years of trying to ignore them, I have come up with this attempt to explain some of the best–known scientific equations, and the various terms that they contain.

I hope it goes without saying that in any equation, you can calculate any one of the values as long as you know all of the others. Or, to put it another way: you can be asked to work out the value of any one of the variables in the following equations – it won't necessarily be the one on the left hand side as I've stated them here.

E=mc²

The most famous scientific equation of them all is part of Einstein's theory of relativity, which tells us that mass and energy are the same physical entity, and can be changed into each other; and that the energy (E) of a body at rest is equal to its mass (m) multiplied by the square of the speed of light (c²).

E	The amount of energy that is equivalent to a body's mass
m	The mass of the body
c	The speed of light

The equivalence of mass and energy was first described as a paradox by the French scientist Henri Poincaré, and proposed by Einstein in 1905.

A question in Macclesfield Quiz League in March 2016 asked which scientific theory, of 1905, was described by this equation. "Relativity" was not enough; it had to be "Special relativity".

This struck me as particularly pedantic. I am aware that Einstein published his theory of relativity in two parts – the special in 1905 and the general in 1915 – but does this equation not form part of both? In trying to nail this, I came across a page on Wikipedia entitled Annus Mirabilis papers. In the course of describing the papers that Einstein published in the year 1905, this page covers both Special Relativity and Mass–energy equivalence – but in separate sections. I'm not enough of an expert to say whether or not the question setter was right in saying that "E=mc2" describes Special Relativity; but I would take a lot of convincing to agree that he or she was right to insist on this exact wording in the answer.

Electrical power

In an electrical circuit, the power that an electric current generates (measured in watts) is equal to the current flowing through the circuit (measured in amperes) multiplied by the potential difference that the current passes through (measured in volts).

This is expressed as P = VI: power (in watts) is equal to the voltage, in volts, multiplied by the current, in amperes (a.k.a. amps).

P	The power, in watts
V	The voltage, in volts
I	The current, in amperes

Electrical resistance: Ohm's Law

Ohm's Law states that the current through a conductor between two points is directly proportional to the voltage across the two points. In other words, the voltage divided by the current is a constant. That constant is known as the resistance of the conductor.

This is expressed as V = IR: the voltage (in volts) is equal to the current (in amperes, a.k.a. amps) multiplied by the resistance (in ohms).

V	The voltage, in volts
I	The current, in amperes
R	The resistance, in ohms

Moving bodies

The velocity (or speed) of a body is calculated as the distance it travels, divided by the time it takes to travel that distance.

This is expressed as v = t/s.

Note that in physics, the convention is (rather confusingly) that s stands for distance and not speed.

It follows from the previous equation (as well as from common sense) that the time a body takes to travel a given distance, at a constant velocity, is equal to the distance divided by the velocity.

This is expressed as t = s/v, or "time equals distance over speed".

If a body is under constant acceleration over a given period of time, its velocity at the end of that time (the final velocity) will be equal to its velocity at the start of the period (the initial velocity) plus the acceleration (in metres per second per second) multiplied by the time (in seconds).

This is expressed as v = u + at.

If a body is under constant acceleration over a given period of time, the distance it travels in that time is equal to its initial velocity (in metres per second) multiplied by the time (in seconds), plus one half of the acceleration (in metres per second per second) multiplied by the square of the time (in seconds).

This is expressed as s = ut + ½at².

v	The final velocity, in metres per second
u	The initial velocity, in metres per second
a	The acceleration, in metres per second per second
t	The time, in seconds
s	The distance travelled

Note that although in everyday speech the terms 'speed' and 'velocity' are interchangeable (and indeed I have used them pretty much interchangeably on this page), in science they are different. The difference is that velocity has both magnitude and direction. In other words, you can't define velocity simply in terms of miles per hour (or feet per second, etc.); you must also state the direction. For example, 'twenty miles per hour in a northerly direction'.

Geometrical Formulae

Sum of the internal angles of a polygon with n sides		(n – 2) * 180
Area of a triangle		Half the base times the height
Area of a circle of radius r		πr² (pi r squared)
Surface area of a sphere of radius r		4πr² (four times pi r squared)
Volume of a sphere of radius r		4/3 * πr³ (four over three times pi r cubed)
Volume of a cylinder, of length l and cross–sectional radius r		πr²l (pi r squared, times l)

Mathematical Formulae

Sum of all the positive integers up to and including n

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