Should formulas in Physics be memorized?

There are formulas which are definitions, which define what terms mean. So for example:

$$ V=IR $$

is a defintion. It is not obvious that it is so, because it is not defining "$V$", the voltage is defined by instruments, nor is it defining current, this is defined by how much charge crosses a surface per unit time. It is defining resistance. The proper way to write it is:

$$ R = {V \over I} $$

But even this is wrong, as current is best thought of as the response to voltage, in a cause-effect sense (although cause/effect is not a fundamental notion in physics, it is fundamental to engineering, and to human thinking). That means--- you set up a voltage, that's the situation, while the current is a response to this voltage, it is the effect once the resister gets to an equilibrium in this voltage.

So the right way is to define the "conductance", $\frac{1}{R}$ and say:

$$ {1 \over R}= {I \over V} $$

in other words, the conductance is the amount of current produced per unit voltage applied. This is the proper definition, and the formula is now internalized. It is defining the conductance properly.

But really, even though it is defining $\frac{1}{R}$, the best way to write it is really

$$ I = (1/R) V $$

meaning, $I$ is proportional to $V$, and the coefficient of proportionality, which is being defined by this equation is $\frac{1}{R}$. All of these are trivially algebraically equivalent, but you need to internalize the idea--- it's a linear relationship with a defined coefficient. The last equation is the one you need to memorize, and not any of the others, because they are wrong.

Once you understand linear relationships with a defined coefficient, that's 70% of the equations you memorize, they are defining linear relationships and defining the coefficients:

$$ \Delta T = {1\over C_P} \delta Q $$

the temperature change is the reciprocal specific heat times the heat absorbed.

$$ F_f = C_f N $$

The friction force is proportional to the normal force (defining coefficient of friction)

$$ Q = CV $$

The charge on a capacitor is proportional to the voltage difference across the two ends (defining capacitance).

Even the granddaddy of all physics equations:

$$ F=ma $$

is a description of the acceleration response to a force, it is a linear relationship which defines the mass.

These you need to commit to memory in the proper way, as they define the coefficient's meaning, so there's nothing to do. But it's no more difficult than learning what the words mean "specific heat, capacitance, resistance", The thing that makes it difficult is only that about 40% of the definitions, due to historical accident, were chosen stupidly, and the reciprocal of the coefficient is the thing that has the name, not the coefficient. This includes even such fundamental things as "temperature", which is really reciprocal coldness, and is defined by this equation:

$$ \Delta S = {1\over T} \delta Q $$

Historically, energy came after entropy, so people defined things the other way. Temperature is also easier to understand.

The definitions and linear relations are really 60% of your equations. Now there are the IDENTITIES, these are things that are not even equations at all, not even definitions, but unit conversions:

$$ E = mc^2 $$

Einsteins mass-energy equivalence

$$ p = \hbar k $$

$$ E = \hbar \omega $$

DeBroglie's momentum/wavenumber relation (somtimes written obtusely as

$$ p \lambda = h $$

$$ E= hf $$

Here, when written properly, the left and right hand sides are things that people once thought were separate things, but are really the same thing once a more fundamental theory is found, except we used different units for the two sides. To get rid of these equations, always, always first learn with a choice of units which makes it that:

$$ c=1 $$

So space and time have the same units, and

$$ \hbar=1 $$

so energy and radian frequency have the same units.

This gets rid of 75% of your equations.

The ones left behind are actual, honest to goodness, physics equations! For example;

$$ PV = NRT $$

The ideal gas law. These equations can be derived from underlying principles, and you need to understand how this works. But the honest truth is that that's only like $1$ equation a week in an elementary physics class, the rest of the time, you are doing nonsense with defining units and working with linear relationships, and learning to deal with annoying reciprocal conventions.

To learn $PV = NRT$, first write it properly $P = n R T$, where $n$ is the density. Then swap out the units so that $R=1$ (first by changing moles for number of atoms, so that $R$ goes to Boltzmann's constant, and then setting Boltzmann's constant to $1$, so that $T$ is in energy units). Then you have

$$ P = nT $$

The pressure is equal to the inverse coldness times the density. Why should the pressure be the inverse coldness times the denisty in a gas? Now you can look at a derivation, from kinetic theory, or from thermodynamics, and understand what parts are important for the derivation.

This is what you need to do, get rid of the nonsense units and definitions, so that you focus on the real content. This is 80% of the first three years of physics education, and the difference in aptitude of students is basically the random process of who gets the trivialities and who doesn't. if you don't understand the trivialities, you have about $10$ to $100$ times the work of someone who does, and it is much more boring also, because you don't understand.