Simulation · Physics · Class 11
Equal areas in equal times
From the lesson The law of areas in Gravitation. Change the values and watch what happens.
Equal areas in equal timesPhysics · Class 11
The idea behind it
NCERT §7.2
- Law of areas: in any two equal stretches of time, the sun–planet line covers the same area. A planet therefore speeds up near the sun and slows down far from it.
- In a short time Δt the sun–planet line sweeps ΔA = ½ |r × v Δt|, so ΔA/Δt = L/(2m), where L = r × p is the planet's angular momentum.
- Gravity on a planet points along the sun–planet line (a central force), so it exerts no torque about the sun; L stays constant and so does ΔA/Δt.
- The law of areas therefore holds for any central force, not only for an inverse-square one.
- At perihelion and aphelion r and v are perpendicular, so m r_P v_P = m r_A v_A, giving v_P / v_A = r_A / r_P: the planet is fastest at perihelion.
- The half of the orbit around the aphelion encloses the larger area, so the planet spends the longer time on that half.