Kepler's LawsJEE Main

Elliptical orbits and equal areas — interactive Physics simulation for IIT-JEE.

Concept

Kepler's three laws: (1) planets move on ellipses with the Sun at one focus; (2) the Sun–planet line sweeps equal areas in equal times (the planet moves fastest at perihelion); (3) $T^2 \propto a^3$ . Law 2 is angular-momentum conservation; law 3 follows from the inverse-square force.

Key formula

T^2 = \frac{4\pi^2}{GM}a^3, \qquad \frac{dA}{dt} = \frac{L}{2m} = const, \qquad v^2 = GM\!\left(\frac{2}{r} - \frac{1}{a}\right)

Derivation

Equal areas: $dA = \tfrac12 |\vec r \times \vec v\,dt| = \frac{L}{2m}dt$ — constant because gravity is central (zero torque).

At the apsides $\vec v \perp \vec r$ , so $mv_pr_p = mv_ar_a$ gives $v_p/v_a = (1+e)/(1-e)$ .

The vis-viva equation comes from energy conservation with $E = -GMm/2a$ .

Scenarios to explore

Kepler's Laws — Elliptical orbit, equal areas in equal times, T² = a³.

Real-world applications

Hohmann transfer orbits between planets.
Comet timing — Halley's 76-year period fixes its a ≈ 17.8 AU.
Exoplanet mass/period determination from stellar wobble.

JEE exam tips

r_p + r_a = 2a and r_p·r_a = b² — two instant identities.
Total energy E = −GMm/2a depends ONLY on a, not on e.
Speed ratio at apsides = (1+e)/(1−e) via angular momentum — no vis-viva needed.

Common mistakes

Placing the Sun at the ellipse's centre instead of a focus.
Using T² = a³ with SI units — it needs AU & years (or the full 4π²/GM).
Assuming constant orbital speed on an ellipse.

Exam traps to avoid

Doubling a multiplies T by 2√2, not 2 (three-halves power).
Equal areas ≠ equal arc lengths — the planet covers more DISTANCE per day at perihelion.