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Many students try to learn the list of formulas for each Class 11 physics chapter by heart. It works until a question changes one detail, such as a ball thrown from a wall instead of the ground, and the memorised formula quietly gives the wrong answer. The alternative is to learn a few laws well and rebuild formulas from them when you need them.
What does learning physics from first principles mean?
It means treating formulas as results, not starting points. You start from a law you trust (Newton's second law, conservation of energy, constant acceleration under gravity), write it down for the situation in front of you, and do the algebra until the quantity you want appears.
This is close to how experts think. Michelene Chi, Paul Feltovich and Robert Glaser found that experts represent physics problems by the principle behind them, while novices go by the problem's literal features (Cognitive Science, 1981). Deriving from laws makes you name the principle first, which is the expert's starting point.
How do you derive a formula instead of memorising it?
Use the same three steps every time:
Pick the laws. Name the principle in words before writing anything.
Derive. Write the law for this situation, with a diagram, axes and signs. Solve symbolically and put numbers in only at the end.
Check. Test the result with units and with limiting cases, values where you already know what must happen.
A derivation with a sign slip often survives a glance but fails a limiting case. Our post on dimensional analysis as a 30-second check goes deeper on units.
Worked example: where does the projectile range formula come from?
A ball is launched from level ground with speed u at angle θ above the horizontal. Ignore air resistance. How far does it land?
Step 1, the laws. No horizontal force acts, so the horizontal velocity stays u cos θ. Vertically, the acceleration is g downward, so the vertical velocity starts at u sin θ and changes steadily.
Step 2, derive. Horizontal position: x = (u cos θ) t. Vertical position: y = (u sin θ) t − ½ g t².
The ball lands when y = 0 again. Factor out t: t (u sin θ − ½ g t) = 0. The answer t = 0 is the launch, so the flight time is t = 2u sin θ / g.
Put that time into the horizontal equation: R = u cos θ × 2u sin θ / g = 2u² sin θ cos θ / g. Using 2 sin θ cos θ = sin 2θ, the range is R = u² sin 2θ / g.
Step 3, check.
Units: u² is in m²/s² and g in m/s², so R comes out in metres.
θ = 0°: fired along the ground, it never leaves it; sin 0 = 0, so R = 0.
θ = 90°: fired straight up, it lands where it started; sin 180° = 0, so R = 0.
Weaker gravity: a smaller g, as on the Moon, should carry the ball further, and g is in the denominator.
With u = 20 m/s, θ = 45° and g = 9.8 m/s², the range is 400 / 9.8, about 41 m. Since sin 2θ cannot exceed 1, 45° gives the largest range on level ground. For more on height and time of flight, see projectile motion on level ground.
Why does this beat memorising the formula?
Launch the same ball from the top of a 10 m wall. It no longer lands at y = 0, so u² sin 2θ / g is wrong, and 45° no longer gives the longest throw. A student who memorised the formula has nothing to fall back on. A student who derived it repeats step 2 with y = −10 m, solves a quadratic for t, and runs the same checks.
How to practise this week
Derive five formulas from your current chapter on a blank page, without the book.
For each, write one units check and two limiting cases.
Change one condition in a textbook problem and solve the new version from the laws.
While a topic is new, mix in worked examples; our summary of worked examples versus solving from scratch explains when each helps.
If you get stuck mid-derivation, the Fermi Problem Library has worked problems with a step-by-step solver that explains the reasoning rather than printing an answer. If a trajectory is easier to see than to read, the Fermi Simulation Lab has interactive simulations for ideas like that. And if you are not sure where to start, Fermi's AI tutor works out what you already know and builds a plan for the session.
