Some time ago, I found myself in California on a ferry to Catalina Island. It was overall an enjoyable experience; the cool, salty air blowing over the rolling waves and late-summer sun were entrancing. At one point we were visited by a pod of dolphins that swam and leapt between the bow waves of our boat. I sat in the shade drinking a beer and enjoying the ride. Like I said, it was enjoyable, but this part isn’t really relevant (just setting the scene here).
On the way back to port, we were navigating through an industrial shipping yard lined with towering cranes, the sort that lift the 40’ metal crates off shipping barges. Two college-age guys plopped down on the barstools next to me. As we passed by the tall cranes, the following conversation took place:
“Dude, what if someone jumped from up there?”
“They’d die, for real!”
“Well like, how deep do you think you’d go if you jumped from up there?”
“12 feet,” (said with a sense of absolute authority)
“No way, you’d go way deeper than that brah!”
“Nope. 12 feet.”
I can’t recall what was said after that because I couldn’t get over the attitude of the guy. 12 feet? Seriously? With no explanation on how or why it couldn’t possibly be any deeper than that? I spent the rest of the ride working the math out in my head, but the equations quickly became too complicated.
I had some time this afternoon, and for some reason I thought back to that conversation. I decided that I’d like to work it out using some of my engineering tools. The following is my thought process:
Stating the problem
I started by stating the question:
What’s the MAXIMUM depth to which a person can dive, as a function of the height from which they jump?
In this context, I’m treating “depth” as the smallest vertical distance from the surface of the water to the nearest point on the diver. I’m also making a few assumptions:
- The diver is a typical human being of average density
- The diver will be treated as a rigid body
- Normal atmospheric conditions at sea level
- The fluid is seawater
- The collision occurs without tumbling
Assumptions (2) and (5) need a bit of explaining. During freefall, the diver’s potential energy is converted to kinetic energy. This is the energy that will propel the diver through the water until the various fluid forces can slow them down to a stop. The more kinetic energy that is retained in the collision with the surface, the deeper the dive will be.
In reality, energy will be lost in the collision by breaking the surface tension, warping the body, and possibly breaking the diver’s insides. This reduces the energy available to propel the diver beneath the surface and results in a shallower dive. Since this problem is looking for the MAXIMUM depth, it’s fair to assume ideal conditions, like a rigid, (reasonably) aerodynamic body that will conserve energy throughout the collision and pass through the water without tumbling.
We can then split the problem into three distinct phases:
- free fall through the atmosphere,
- transitioning from air to water (collision),
- slowing through water until velocity is zero.
In any given phase, the major forces we’re dealing with are gravity (down), buoyancy (up) and drag (opposite direction of travel).
We know that the speed of a diver before hitting the surface of the water is limited by the drag force; the maximum speed he can reach is his terminal velocity. Put another way, higher dives will translate into faster entry speeds, but only to a point. Beyond this point, increasing the altitude of the dive will not lead to a higher entry speed. Since we’re assuming negligible energy loss at entry, for the split second that the diver is transitioning from the air to the water, he would have 4 major forces to deal with: gravity, two drag forces (air and water), bouyant forces (water only, we’re neglecting air). Finally, when he’s completely submerged, the forces due to air are gone, and he’s back to gravity, drag and buoyancy. In a fluid, these are typically more substantial than in air, so at some point they will slow his velocity to zero, before forcing him back to the surface. The depth he’s at when this occurs is his maximum depth.
So we have a rough model for how deep the dive should be, but what about the numbers? To get there, let’s look at the equations:
(Free falling)
(Entering the water)
(Under the water)
Equations 1 and 3 are actually the same form. The only difference between the two is the presence of the buoyant force in 3 (it exists, but is disregarded as negligible in 1). These equations can be generalized as:
One solution to this form is:
Incidentally, with an initial velocity of 0, an initial displacement of 0, a sufficiently large final displacement, and letting A=g and B=(density x area x drag)/(2 x mass), the above equation provides a solution for Vf (treating “downward” motion as positive):
…which is the terminal velocity of an object in freefall without buoyancy.
Equation 2 is a bit trickier. I don’t have any classical tools that will let me solve for y directly, so I’ll need to use an algorithm. I only need to start with the initial position and initial velocity, which in this case are the components of motion in effect when the jumper makes contact with the water. Plugging in these initial values allows me to find the initial acceleration. I can then make a small “step” in time, say .01s, and estimate the position and velocity at this time. Plugging these values into Equation 2 allows me to estimate the acceleration at this point in time. Following this process recursively until the diver is completely submerged will allow me to model his speed and position throughout the transition from air to water.
Since I need a script to complete this portion, and I am curious how the outcome would be affected by tweaking one or more variables, I decided to write a MATLAB/Octave simulation. I researched reasonable ranges for things like: drag coefficients and reference areas (Olympic swimmer to skydiver); mass (95% of U.S. adults are between 45 and 95kg); human density, etc. I even iterated over ridiculous jump heights (380m - the height of the Empire State Building). Enough to reach terminal velocity.
The result was definitely interesting: after I checked/re-checked the code and set up arrays of realistic values for our variables, the absolute maximum depth computed in 157,248 scenarios is 4.6 meters - a little more than 12 fucking feet. Since I didn’t take into account for things like breaking the surface tension of the water or the energy of a body crumpling/warping on impact, this number is actually a little higher than would be expected in practice. So yeah, 12 feet is actually a reasonable assertion.