# Good ways for learning and cramming formulas? [duplicate]

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?

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## marked as duplicate by jinawee, Qmechanic♦Mar 11 at 11:27

I'm afraid that for the most part we learned them by deriving them. Always know the reason; never just cram the formulas. If you do you might pass the exam, but you'll find it hard to make progress afterwards. Maybe not so much help to you now, but you asked how we learnt all those formulas, and for most of us that's how it is. –  Nathaniel Mar 11 at 10:30
no, I didn't cram any formulas(maybe some) .I know how to derive them and can derive them at any point but in exam its not viable.what I am asking if I want learn them fast ,how can i do it? –  user4678 Mar 11 at 10:34
This question appears to be off-topic because it is about studying techniques. –  jinawee Mar 11 at 10:52
IMO formulas are best learned through osmosis. Not just derivation, but repetitive usage. –  Stan Liou Mar 11 at 11:15
@user4678 if you practice deriving them enough, you should be able to do it in your head in seconds. It's not about writing out the full derivation, it's about having such a clear understanding of where the formula comes form that the formula itself just becomes obvious. As Stan Liou says, just using them will help with this as well. Sooner or later they will just become second nature. –  Nathaniel Mar 11 at 11:16

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 $$Joules = kg \cdot \frac{m^2}{s^2}$$ 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: $$E.K. \propto m v ^2$$ 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.

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You can use formulas you know to work out the units as well, i.e. $E=mc^2$ So $[J] = [kg][ms^{-1}]^2 = [kg][m^2][s^{-2}]$ –  user288447 Mar 11 at 11:13

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 $C$.

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