Why Do Planets Speed Up Near the Sun? Kepler's Second Law

A planet whips around the Sun when it is close and crawls when it is far away. Yet the line from the Sun to the planet sweeps out the same area in the same time. Why would nature keep that area the same? This video builds the answer from three simple ideas, ellipses, triangle areas and motion, and then follows Isaac Newton's proof using nothing but triangles. 00:00 Why the same area? 00:46 Copernicus, Brahe and Kepler 01:51 Ellipses and where the Sun sits 02:48 Triangle area: slide the corner 04:20 Motion: steps and tugs 05:03 Newton's triangles 06:53 A tug after every step 08:03 Why faster up close? 09:55 Other pulls, and Earth's calendar 12:00 Summary Copernicus pictured the planets moving on circles at a steady speed. Tycho Brahe's precise positions for Mars would not fit a circle, and Johannes Kepler found that the orbit is an ellipse with the Sun off to one side, and that the planet's speed is not steady. He wrote the area rule in its modern form in 1621, and in 1684 Newton proved it with his laws of motion. Why it works: a triangle's area is half its base times its height, and any side can be the base. With no pull, an object moves in a straight line by equal steps, so the triangles it sweeps from the Sun have the same base and height and the same area. A tug toward the Sun shifts the next point parallel to the line from the object to the Sun, so the corner slides parallel to that base, the height stays the same, and so does the area, whatever the tug's strength. Tug after every step and the jagged path becomes the real orbit, still sweeping equal areas in equal times. Why faster up close: a thin slice is almost a triangle with base equal to the distance and height about the distance times the angle, so its area is about one half times the distance squared times the angle. Equal times give equal areas, so where the distance is small the planet must turn through a wider angle. In the drawing the near point is 4 times closer, so the planet turns 16 times faster there and moves 4 times faster along its path. The proof only used where the pull points. A spring-like pull toward a center also sweeps equal areas, but a push that does not point to the center breaks the rule. Physicists know this rule as the conservation of angular momentum. Earth's orbit is almost a circle. Earth is closest to the Sun around January 3 and farthest around July 4, moving at about 30.29 km/s and 29.29 km/s. From the March equinox to the September equinox takes about 186 days, and back to March about 179 days. The orbit in most of the video is stretched for clarity (eccentricity 0.6); Earth's is about 0.017. Watch the Short version:    • Kepler's Second Law: Why Planets Speed Up ...   References: Kepler's laws of planetary motion (Wikipedia): https://en.wikipedia.org/wiki/Kepler%... Areal velocity (Wikipedia): https://en.wikipedia.org/wiki/Areal_v... Earth's orbit (Wikipedia): https://en.wikipedia.org/wiki/Earth%2... Isaac Newton, Principia, Book I, Section II, Proposition I (Motte translation, 1846): https://en.wikisource.org/wiki/The_Ma... Narration: AI-generated voice. Seeing Is Proof — big ideas, proven visually. #physics #math #kepler