Did you get it right?
➡ The Problem
The dastardly Sheriff of Nottingham plans to rally his troops in his fortress, protected by a large moat with only a single drawbridge. As the drawbridge begins to rise behind the Sheriff’s troops, a lone figure appears on the field before the castle—the one and only Robin Hood!
The legendary archer’s only chance to save the people of Sherwood Forest is to race toward the fortress on his faithful steed, leap onto the bridge before it can close, and disrupt the rally before it can even begin. With his incredible tactical acumen, Robin Hood surveys the field and begins to make a plan.
The drawbridge is 32 feet long. It’s already raised 2 feet above the ground and seems to be lifting at a constant 2 degrees per second. Robin Hood is 200 feetaway from the edge of the drawbridge. Without hesitating, he spurs his horse into a gallop and attempts to make the jump before it’s too late.
So, how does this story end? Does Robin Hood make it onto the bridge? Or does he end up pulling himself out of the moat?
➡ The Solution
This problem requires us to compare two things: the physical ability of Robin Hood’s horse and the height of the bridge as time passes. One of these asks for estimation while the other can be calculated directly.
Like any creature, horses that are trained to jump will build muscle that lets them jump significantly better than their peers. Typical show horses start jumping over fences set anywhere from 2.5 to 3 feet, with a world record sitting at just over 8 feet.
In terms of galloping speed, the average horse clocks in between 25 and 30 mph, or roughly 40 feet per second. Let’s assume that the horse can gallop and leap without meaningfully changing speed. That means that an average horse will cross the 200 foot field to the bridge in 5 seconds.
So our new task is to find out how high the bridge is after 5 seconds. We know it starts at 2 feet and that the angle the far side of the drawbridge makes with the ground is increasing at 2 degrees per second. After 5 seconds, we’ve added 10 degrees. What angle did it start at? Let’s sketch it out.
Let’s call the angle a. When we set up this problem with a right triangle, we see that the easiest path to a solution involves some basic trigonometry. The sine ratio compares an angle to the opposite side divided by the hypotenuse.
So, a horse galloping at average speed would need almost a world record leap to make it over the bridge and into the castle! But didn’t we say Robin Hood likely had a faster horse? Let’s consider three mounts: the Jumper, the Cannon, and the Faithful Steed.
The leap isn’t impossible, but it will take a trained horse that is above average in terms of both speed and jumping ability. Our Faithful Steed was closest, and they’ll actually clear the jump if they just run a bit faster—63.3 ft/sec or about 43.2 mph. A small but important increase for the likes of the legendary Robin Hood!