Essential Mathematics

This section describes some of the mathematical techniques that are needed in studying A-Level Physics.

Quantities and units
Physical quantities are described by the appropriate words or symbols, for example the symbol R is used as shorthand for the value of a resistance. The quantity that the word or symbol represents has both a numerical value and a unit e.g. 10.5 Ω . When writing data in a table or plotting a graph, only the numerical values are entered or plotted.

For this reason headings used in tables and labels on graph axes are always written as (physical quantity)/(unit), where the slash represents division. When a physical quantity is divided by its unit, the result is the numerical value of the quantity.

Resistance is an example of a derived quantity and the ohm is a derived unit. This means that they are defined in terms of other quantities and units. All derived quantities and units can be expressed in terms of the seven base quantities and units that make up the SI, or International System of Units.

The quantities, their units and symbols are shown in the table. The candela is not used in A Level Physics.

Equations
The equations that you use in Physics are relationships between physical quantities. Since it is not possible for one quantity to equate to a different quantity, an equation must be homogenous, i.e. the units on each side of the equation must be the same. This is useful for:

finding the units of a constant such as resistivity
checking the possible correctness of an equation; if the units on each side are the same the equation may be correct but if they are different it is definitely wrong.

It is important to distinguish between a physical quantity and its unit. The value of a physical quantity includes both the numerical value and the unit it is measured in.

When working out the units of an equation, make it clear that you are dealing with units only by writing units of (quantity) = (unit).

Often equations need to be used in a different form to that in which they are given or memorised. The equation needs to be rearranged. When rearranging an equation it is important to remember that:

the same mathematical operation must be applied to each side of the equation
addition or subtraction should be done first, followed by multiplication or division and finally roots and powers.

For example, to rearrange the equation v² = u² + 2as to enable u to be calculated:
1. subtract 2as from each side of the equation to give v² – 2as = u²
2 .square root each side of the equation to give U(v2 – 2as) = u

An equation that relates the quantity to the product or quotient of two others can be rearranged using the ‘magic triangle’, a triangle divided into three.

To use the triangle, start with the quantities that are multiplied or divided in the relationship. For example, in V = IR, you would write I and R into the triangle first. Two quantities that are multiplied together are written side-by-side at the bottom of the triangle. Where one quantity is divided by the other these are written into the triangle as they would be in a division, with one quantity ‘over the other’.

The remaining quantity is now written into the vacant slot. This is illustrated below.

To use the triangle to make I the subject of the equation, cover up the I and you are left with ‘V over R’, so I = V/R.

Drawing graphs
Graphs have a number of uses in Physics:

they give an immediate, visual display of the relationship between physical quantities
they enable the values of quantities to be determined
they can be used to confirm or disprove a hypothesis about the relationship between variables.

When plotting a graph, it is important to remember that values determined by experiment are not exact. They are subject to both the accuracy and precision of any measuring instrument and the person using it, as well as any changes that the measuring instrument itself might cause. Consider measuring the temperature of a liquid with a simple thermometer.

The manufacturer specifies the accuracy, the extent to which the thermometer is reliable. Typically, on a standard laboratory thermometer this is ±1°C.
When taking a reading, the user may read to the nearest half degree or nearest degree. This describes the precision.
Inserting the thermometer into a substance may change the temperature of the substance as the two reach thermal equilibrium.

For these reasons, having plotted experimental values on a grid, the graph line is drawn as the best straight line or smooth curve that represents the points. Where there are ‘anomalous’ results, i.e. points that do not fit the straight line or curve, these should always be checked. If in doubt, ignore them but do add a note in your experimental work to explain why you have ignored them and suggest how any anomalous results could have arisen.

Measurements and graphs
Quantities such as velocity, acceleration and power are defined in terms of a rate of change of another quantity with time. This rate of change can be determined by calculating the gradient of an appropriate graph. For example, velocity is the rate of change of displacement with time. Its value is represented by the gradient of a displacement-time graph. Different techniques are used to determine the gradient of a straight line and a smooth curve. For a straight line:

determine the value of Δy, the change in the value of the quantity plotted on the y-axis, using the whole of the straight line part of the graph
determine the corresponding value of Δx
calculate the gradient as Δy ÷ Δx

For a smooth curve, the gradient is calculated by first drawing a tangent to the curve and then using the above method to determine the gradient of the tangent.

To draw a tangent to a curve:

mark the point on the curve where the gradient is to be determined
use a compass to mark in two points on the curve, close to and equidistant from the point where the gradient is to be determined
join these points with a ruler and extend the line beyond each point.

These techniques are illustrated in the diagrams below.

The area between a graph line and the horizontal axis, often referred to as the ‘area under the graph’ can also yield useful information. This is the case when the product of the quantities plotted on the axes represents another physical quantity, for example, on a speed-time graph this area represents the distance travelled. In the case of a straight line, the area can be calculated as that of the appropriate geometric figure. Where the graph line is curved, then the method of ‘counting
squares’ is used.

Count the number of complete squares between the graph line and the horizontal axis.
Fractions of squares are counted as ‘1’ if half the square or more is under the line, otherwise ‘0’.
To work out the physical significance of each square, multiply together the quantities represented by one grid division on each axis.

Equations from graphs
By plotting the values of two variable quantities on a suitable graph, it may be possible to determine the relationship between the variables. This is straightforward when the graph is a straight line, since all straight line graphs have an equation of the form y = mx + c, where m is the gradient of the graph and c is the value of y when x is zero, i.e. the intercept on the y-axis. The relationship between the variables is determined by finding the values of m and c.

The straight line graph below shows how the displacement, s, of an object varies with time, t.

Trigonometry and Pythagoras
In the right-angled triangle shown below, the sides are labelled o (opposite), a (adjacent) and h (hypotenuse). The relationships between the size of the angle θ and the lengths of these sides are:

sin θ = o/h
cos θ = a/h
tan θ = o/a

Pythagoras’ theorem gives the relationship between the sides of a right-angled triangle: h² = a² + o²

Multiples and submultiples
For AS Physics, you are expected to be familiar with the following multiples of units:

For example, the symbol MHz means 1x106 Hz and mN means 1x10-3 N.

Equations that you need to know

This is a list of equations that may be needed but are not provided in topic tests or end-of-course examinations. Not all of these equations are needed for every specification as some are in the A Level and not the AS specification. The table shows the equations needed for each AS specification.
All other equations that are needed will be provided on a data sheet, along with any data such as the values of physical constants.

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