Kepler's Second Law: A Planet's Orbit Sweeps out Equal Areas in Equal Times

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In this video, I derive Kepler's 2nd law of planetary motion and show that the line joining the Sun to a planet sweeps out equal areas in equal times. In other words, the rate of change of the area of the orbit being swept is constant in time. Using the result from my earlier proof of Kepler's first law, where I showed that r x v = h = a constant vector, I first evaluate this integral to obtain an equation for h. I then take a thin slice of the planet's orbital ellipse and approximate it as the area of a sector of a circle. This allows us to write the area being swept by the planet's orbit as an integral, whose integrand is thus the derivative of the area, as per the Fundamental Theorem of Calculus. Since the integrand is just h/2, this is a constant value, thus proving Kepler's 2nd law, that is, the rate of change of the orbital area being swept is constant.

#math #vectors #calculus #ellipse #keplerslaws

Timestamps

• Applied Project: Kepler's three laws – 0:00
• Problem 1: Prove Kepler's Second Law – 1:20
• Solution: Write position vector in polar coordinates – 2:45
• Solution to (a): Use the proof of Law 1 to obtain a formula for the h vector – 4:46
  • Recap on the Pythagorean Trigonometric Identity – 14:43
• Solution to (b): Deduce the constant magnitude of the h vector – 15:46
• Solution to (c): Derivative of the area of orbit swept in time – 19:06
  • Recap on the area of a sector of a circle – 20:17
  • Draw a thin sector of the ellipse and obtain an integral for the area being swept – 21:11
  • Derivative of the area is the integrand of the integral – 24:56
  • Recap on the Fundamental Theorem of Calculus – 25:47
• Solution to (d): Since h is a constant vector, the derivative of the area is constant – 27:36
  • This proves Kepler's 2nd Law – 30:01

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