Good ways for learning and cramming formulas?

Question

I got an exam coming up and its numerical based (its pre-university level exam) but really tough. I want to know about various ways and methods to learn formulas. I know how can I derive them but its time consuming. I want to know how you people learnt these formulas?

Answer 1

The best people way to learn formulas is to know the units for all the different quantities. You can then figure out almost any formula you want by reasoning it out. As a simple example consider the kinetic energy formula. The units of energy are \begin{equation} Joules = kg \cdot \frac{m^2}{s^2} \end{equation} If you remember that kinetic energy depends on the mass and velocity(or you can just reason what kinetic energy could possibly depend on...) then there is only one option: \begin{equation} E.K. \propto m v ^2 \end{equation} At this point you do need to memorize the factor of $1/2$ in front but units got you almost the entire way there.

This idea is extremely helpful is checking to make sure you remember your formulas correctly since if the units on the left and right hand side can't be converted into one another then you know you got your formula wrong.

Answer 2

A great tool to quickly derive a formula for a quantity is dimensional analysis. Essentially, you identify the dimensions, or units, of all relevant quantites, and derive the formula for another quantity, e.g. energy by combining them in such a way that the dimensions are correct.

A famous example of the power of this formalism is given by physicist G.I. Taylor in the 1950s. He wanted to compute the energy released of an atomic explosion. He identified the relevant variables:

Shock front radius [R] = length; time from explosion [T] = time; and air density [$\rho$] = mass per length$^3$. Since the energy has dimensions of mass times length$^2$ times $\mathrm{time}^{-2}$, he deduced:

$$E = C \frac{\rho R^5}{t^2}$$

up to a dimensionless constant $C$.