PhysicsGravitationClass 11

Gravitation and orbits: interactive simulation

The same inverse-square force that drops an apple keeps the Moon in orbit and the planets on their ellipses. Kepler found the patterns; Newton showed they all follow from one law.

Useful for: JEE Main, JEE Advanced, NEET, CBSE Boards, AP Physics 1, AP Physics C: Mechanics, A-level Physics, IB Physics · 4 modes in one lab

What you can explore

  1. 1

    Kepler's laws

    Elliptical orbits, equal areas in equal times, perihelion and aphelion speeds and T² ∝ a³.
  2. 2

    Orbit & escape speed

    Newton’s cannon from sub-orbital to circular, elliptical, parabolic and hyperbolic paths.
  3. 3

    Variation of g

    g against distance from the centre, inside and outside the Earth, plus the effect of latitude.
  4. 4

    Tunnel through Earth

    Simple harmonic motion through a straight tunnel and its 84.6-minute period.

Try this

Change an orbit’s eccentricity and watch the equal areas, fire a cannonball faster and faster until it never returns, find g at any height or depth, and see why a hole through the Earth would make a 42-minute train ride.

  1. 1

    Set e = 0.6 in Kepler’s laws.

    What you should see

    The planet is 4 times faster at perihelion than at aphelion, (1 + e)/(1 − e), yet every shaded sector has the same area.

  2. 2

    Double the semi-major axis from 1 to 2 AU.

    What you should see

    The period becomes 2^1.5 ≈ 2.83 years, independent of e.

  3. 3

    In Orbit & escape speed, try 0.8, 1, 1.25 and √2 times 7.9 km/s.

    What you should see

    Falls back, circles, traces an ellipse, then escapes along a parabola. Total energy changes sign at √2.

  4. 4

    In Variation of g, go 1000 km up, then 3200 km down.

    What you should see

    g ≈ 7.33 m/s² at 1000 km; 4.9 m/s² halfway to the centre; 0 at the centre.

  5. 5

    In Tunnel through Earth, try a tunnel that misses the centre.

    What you should see

    The period stays 84.6 minutes (R = 6400 km); only the maximum speed changes.

Formulas behind it

Newton’s law of gravitation

F=GMmr2F = \frac{GMm}{r^2}

Kepler’s third law

T2∝a3T^2 \propto a^3

Orbital and escape speed at the surface

vo=gR≈7.9 km/s,ve=2gR≈11.2 km/sv_o = \sqrt{gR} \approx 7.9\ \text{km/s},\qquad v_e = \sqrt{2gR} \approx 11.2\ \text{km/s}

g at height h and depth d

gh=g(1+h/R)2,gd=g(1−dR)g_h = \frac{g}{(1 + h/R)^2},\qquad g_d = g\left(1 - \frac{d}{R}\right)

Effect of Earth’s rotation

g′=g−ω2Rcos⁡2λg' = g - \omega^2R\cos^2\lambda

Period of a tunnel through the Earth

T=2πRg≈84.6 minT = 2\pi\sqrt{\frac{R}{g}} \approx 84.6\ \text{min}

Common mistakes

  • Using g(1 − 2h/R) for large heights. It is only an approximation for h ≪ R.
  • Thinking escape speed depends on the mass of the object or the launch direction (ignoring air).
  • Saying there is no gravity on the ISS. g there is still about 90 % of its surface value; astronauts are in free fall.
  • Taking the planet’s speed as constant on an ellipse.
Practise 223 questionsCheck your answer instantly, with a worked solution.Stuck on a gravitation and orbits question?Snap it and get a whiteboard video solution.

FAQ

Why is g larger at the poles than at the equator?

Two reasons: the Earth is slightly flattened, so the poles are closer to the centre, and at the equator part of gravity provides the centripetal force for Earth’s rotation (ω²R ≈ 0.034 m/s²).

What does Kepler’s second law tell us physically?

Equal areas in equal times is the same as saying the planet’s angular momentum about the Sun is constant, because gravity points straight at the Sun and exerts no torque.

Why is a geostationary satellite always at about 36 000 km?

Its period must equal 24 hours, and Kepler’s third law fixes the one orbital radius that has that period: about 42 000 km from Earth’s centre.

Related simulations

All 65 simulations
Projectile motionPhysics, Class 11Simple pendulumPhysics, Class 11Standing waves on a stringPhysics, Class 11Doppler effect in sound & sonic boomPhysics, Class 11 & Class 12Rotational dynamics and rollingPhysics, Class 11Laws of motion, friction and pulleysPhysics, Class 11

Want a specific topic or simulation added?

Tell us in our feedback box which chapter or concept you want visualised next.

Send feedback