Quantities and units in mechanicsAQA A-Level Maths: Revision notes

All mechanics quantities are built from three fundamental (base) quantities, each with an SI unit:

• length: the metre, m
• time: the second, s
• mass: the kilogram, kg Other units are multiples of these, such as km, minutes, hours and grams. Always convert to SI before substituting into a formula, unless every quantity is in a consistent set of units.

A derived quantity is built from the fundamental ones.

• velocity $=\frac{\text{displacement}}{\text{time}}$, unit m s$^{-1}$
• acceleration $=\frac{\text{change in velocity}}{\text{time}}$, unit m s$^{-2}$
• force: newton, N, where $1$ N $=1$ kg m s$^{-2}$
• weight: a force, so measured in newtons Example: speeding up from $5$ m s$^{-1}$ to $8$ m s$^{-1}$ in $6$ s gives acceleration $\frac{8-5}{6}=0.5$ m s$^{-2}$.

Mass is the amount of matter in an object, in kg, and is the same everywhere. Weight is the force of gravity on it, in newtons: $W=mg,$ where $g$ is the acceleration due to gravity. On Earth use $g=9.8$ m s$^{-2}$ unless told otherwise. A $15$ kg crate weighs $15\times9.8=147$ N on Earth, but only $15\times1.6=24$ N on the Moon, where $g=1.6$ m s$^{-2}$. Its mass is $15$ kg in both places. With the newton, Newton's second law is $F=ma$, with $F$ in N, $m$ in kg and $a$ in m s$^{-2}$.

Convert before calculating.

• km h$^{-1}$ to m s$^{-1}$: multiply by $\frac{1000}{3600}$, that is divide by $3.6$.
• m s$^{-1}$ to km h$^{-1}$: multiply by $3.6$.
• minutes to seconds: multiply by $60$; kilometres to metres: multiply by $1000$. Example: $1.2$ km in $4$ minutes is $1200$ m in $240$ s, so speed $=5$ m s$^{-1}$, and $5\times3.6=18$ km h$^{-1}$.

Units can be used to check a formula. Replace each quantity by its unit and simplify. For $T=\sqrt{\frac{2h}{g}}$: $\frac{\text{m}}{\text{m s}^{-2}}=\text{s}^2$, and the square root has unit s, which is the unit of time, so the formula is consistent. Also check that terms being added or equated have the same unit, and that a unit such as $\frac{\text{N}}{\text{kg}}=\frac{\text{kg m s}^{-2}}{\text{kg}}=\text{m s}^{-2}$ is an acceleration.