When a rocket launches with a payload (like a satellite), it needs a fairing. The fairing, essentially the nose cone of the rocket, is the payload cover that aerodynamically makes the spacecraft fly through the earth's atmosphere. But as soon as the rocket has left most of the air behind it, it no longer needs a fairing – it becomes additional weight. The disguise is ejected and falls back to earth.
Now you enter SpaceX, the rocket company founded by Elon Musk, who likes to try new things. It turns out that these panels are not cheap. For the Falcon Heavy rocket, they can apparently cost about $ 6 million. If you can save the disguises instead of simply throwing them away, this can be a big money. And of course, saving money means that it will be cheaper (in the long run) to get into orbit.
So SpaceX is planning to save its disguises. Once the panels are ejected (they are delivered in two parts), they use small engines to guide them to a landing zone. When they reach a sufficiently low height, the panels use a large parachute to slow them even more. Next comes the magic part. The parachute disguise is captured by a large (and fast) boat with a huge net over it. It seems like a crazy plan that a 4th grade student came up with ̵
This is so crazy event, it's a great inspiration for a physics problem. Such an exercise was probably the first step in the project to catch the disguises. Using some rough estimates, we can see if capturing the panel is even possible. That's the goal of a "Back of the Envelope" calculation – a real thing. It's called "the back of the envelope" because you do not even spend enough time on the problem to bother to get a real piece of paper.
Let us do it.
First, how long would it take for a fairing to fall off the rocket? Suppose the disguise falls from a height of 100 kilometers (a rough approach to the edge of space). Of course, the speed of the fairing would increase as it falls, but if it enters the higher density air (at lower altitudes) it would also encounter an air resistance force that increases with speed. Finally, the fairing would reach a constant fall rate at which the downward gravitational force and upward air resistance cancel each other out. This is known as the final speed. If the air resistance depends on the square of the velocity and the shape and mass of the object, it can be calculated as follows (here a recent post with further details on the final velocity):
In this expression we use the following variables:
- m is the mass of the panel. Based on this Wikipedia page, the mass of a panel amounts to 850 kg.
- g is the gravitational field with a value of about 9.8 Newton per kilogram.
- C is the drag coefficient. This value depends on the shape of the object. I will estimate this at about 0.5. Here is one (list of resistance coefficients).
- A is the cross-sectional area (in the direction of movement). The panel is rectangular (seen from the side) with approximate dimensions of 13.2 m × 5.2 m for an area of approximately 68 m 2 .
- Finally, there is ρ. This is the air density at about 1.2 kg / m 3 (at least near the earth).
Together this results in a final speed of 20 m / s You can see my calculations here. If the fairing starts from a height of 100 km, it would take 82 minutes to reach the surface. But I think that's too long. Of course, for much of the time it falls back to Earth, the fairing would run much faster than this calculated final speed, since the air density is so low. As a rough estimate, we assume that the panel takes 41 minutes to get back to the surface. Do not worry, rough estimates are fine.
Now that I have the falling time, I can think of Mrs. Tree. I know it's a "fast" boat – but I really do not know what that means. It's a bit big so it may be fast like 25 mph and not 50 mph. Let's just go with a boat speed of 30 miles per hour. I totally advise here. So, if the boat is traveling at 48 km / h and has 41 minutes to get to the disguise meeting point, what is the target area you need?
In this case, it may be easier to use a time interval of 41/60 hours (since the speed is in miles per hour). During this time interval, we can calculate the straight-line trajectory as follows:
If you imagine that Ms. Tree could travel that stretch in any direction, you see that the entire target surface is a circle. The area of this circle would be 928 square miles. That seems pretty big – but that's still the ocean we're talking about. And this target area could be even smaller. I assume that it takes some time to determine exactly where a falling fairing is and how its trajectory would go.
Now for a physical problem. Really, all these assessments were just an excuse to do some physics. Here is the problem.
A SpaceX fairing falls from a height of 50 km and a constant top speed of 20 m / s. You are the captain of the speedboat Ms. Tree. Mission Control has just determined that the panel will land 12.3 km from your location. As you try to impress everyone, you decide to wait until the last possible moment to reach the meeting place. How long should you wait?
Do you see? Real physical problems. I'm just having fun with the real thing – but it's still fun. OK, but how do you solve that? Of course you could set up some basic algebraic equations and figure out this problem. Honestly, the solution is not too complicated. But I like to do things differently. How about solving Python? I do not know anything about you, but I love Python. It is the best.
Here is the code for my solution and I will give some hints below. You can view the code by clicking on the "pen" button. If you change something, just click "Play" to run it again.
Students (and other people) are often intimidated by coding solutions. The cool thing is that you can solve this problem in several ways. Do not think that you need an elegant algorithm or something. Just do something that works. This is how my method works.
- The key is to divide the problem into small time steps. In this case, I use a time interval of 0.1 seconds. In the end, I have a whole bunch of simple problems – and that's fine.
- I start by modeling the position of the fairing and the boat. Both move at a constant speed and I move each object step by step in a time interval step.
- Note that the boat is not waiting. It starts to move immediately. You may think that is fraud, but it is a simple solution. I will move the boat until it reaches the meeting point and then stop it. In this way, I can count the remaining time to impact and use it as my delay time. The two lines in the diagram are the positions of the two objects. That is, the vertical axis is both the position along the water (in the x direction) and the vertical position. This is a kind of scam, but it gives me an idea of how the objects are moving. You should try it. How about this – here are some homework problems that you can practice with.
- Suppose the captain wants to drive at constant speed to the fairing location (same distance as the previous problem). At what speed should the boat go there on time?
- The disguise has a parachute. Suppose the captain wants to wait as long as possible before leaving for the rendezvous. In this case, however, the cladding drops at a constant speed of 20 m / s until it reaches a height of 2000 meters. At this time, the parachute is used on the fairing at a vertical speed of 4 m / s.
- What is the new target area for parachute disguise in both square miles and square kilometers?  More Great WIRED Stories