It’s ridiculously early on a cold January morning. I’m shuffling my feet to keep warm and to burn off some of my nervous energy. As I hold my breath, my teenage daughter trusts her life to her horse and her training, as they dart around the jumper ring at top speed. The goal is to finish the course of about a dozen jumps with the fastest time, leaving all the rails in place. I know that if she also trusts the physics, she has a good chance at a clean round.
When learning to jump, her (amazing!) coach instilled in her the golden rule: your takeoff point should be the same distance from the base as the rail is above the ground. In physics terminology, her coach was telling her the golden rule of projectile motion – your projectile (i.e. your horse) will go the furthest before landing if you launch at 45° to the horizontal, no matter what speed you are travelling when you take off.
The speed matters, of course, because it will affect how far you go before landing, as well as how high you go at the top of your flight over the jump. But, as golden rules go, it’s an easy one to remember and is solidly based in physics principles.
Is a horse going over a jump really a projectile like an inanimate football? To check this, I loaded a video of Hannah and Ella into some pretty cool (and free) software to analyze the motion of objects (https://physlets.org/tracker/), to see what it looks like from a physicist’s perspective. It turns out that when we plot Ella’s vertical position as a function of horizontal position, it nicely tracks a curve we call a parabola – which is EXACTLY what you expect for any projectile (football, golf ball, tennis ball, flying squirrel or jumping horse).
As you can see in this screenshot from the associated video, our pair took off at an angle of about 35° from the horizontal. With some great helpers, I measured the distance from her launch point to the base of the jump to be about 55 cm and the distance from the base to her landing point at about 60 cm. This matches the graph above from the video analysis, which shows that their jump was not exactly symmetric over the rail (the x = 0 position) – coming in a little closer to the base than where they landed – but pretty good!
Let’s assume then that we can apply equations of projectile motion that we use in first-year physics to Ella’s path. Since we know the launch angle of 35° and the total distance from launch to landing of 115 cm, we can figure out that the launch speed must have been about 3.5 m/s (~12 km/h), which is consistent with a steady canter pace. If Hannah and Ella want to jump higher and/or further before landing, that launch speed needs to go up, but the golden rule of taking off as far from the jump as it is high still applies for maximum distance for their efforts.
Now if we could just get horsey folk to call horizontal rails “horizontal” instead of “vertical”, we’d all be talking the same language!