Catenary vs Parabola

Simon has been studying the differences between a catenary (the curve a free hanging chain makes) and a parabola (the trajectory of a thrown ball), via the SingingBanana math channel. He told me a lot of interesting facts about the catenary:

• it’s the fastest track for any round object to roll to any point on a curve;
• when you roll two balls on a catenary at different heights, they will crash exactly in the middle;
• when turned upside down, a catenary makes the strongest bridge;