SpaceX recently tested the Dragon capsule abort system. The basic idea is to get the capsule off the rest of the rocket in the event of an emergency. The capsule has several rockets that can be fired to lift it away to safety. Of course you want to test this system before you actually have to use it. So you get this awesome video.

Video Analysis of Lift Off

If you ever want to set a trap to ambush me, you should use some type of video like this. It has something cool (SpaceX is awesome) and leaves some interesting questions (like the acceleration). Better yet, it is a video that lends itself to analysis. The camera doesn’t move and the motion of the object is mostly perpendicular to the view.

The first thing I need for a video analysis is to set the scale of the scene. Different objects in the video are different distances from the camera so that the only object I can use is the Dragon capsule itself. According to SpaceX.com, the Dragon has a trunk diameter of 3.7 meters. Now I can use Tracker Video Analysis to mark the location of the capsule in each frame during the abort test.

Ok, you might already have a complaint. You might say “but you can barely see the capsule in this video. How can you use its diameter to set the scale?” That is a great point. I think you are right in that this measurement is possibly off. Let’s just make our best guess and then deal with the uncertainty afterwards.

Here is a plot of the vertical position of the capsule after the rockets fired.

I’m quite surprised that this position data looks like a quadratic function would fit quite well. This would indicate that the vertical acceleration is nearly constant with a value that is twice the coefficient in front of the t2 term giving it a value of 36.6 m/s2. If I estimate the uncertainty in the position measurement is +/- 0.5 meters, this would put the acceleration at 36.6 +/- 4.9 m/s2 (I left off the uncertainty details, you can see how that’s done here). Oh, that would be 3.7 +/- 0.5 g’s.

A Homework Example

This is the part where I would normally list some homework questions for everyone to work on. However, I think I will post one question as an example – just to show you how to do it.

Question: Based on this clip, how high does the capsule go?

I am going to start off with some assumptions.

The capsule starts from rest (that seems obvious).

For the data in the plot above, the capsule goes out of the frame of the camera at around 10.9 seconds. I’m not sure when the rockets turn off, but at around 14 seconds the camera once again shows the capsule with the thrusters off. I’m just going to assume the rockets turn off at 10.9 seconds.

I will assume a constant vertical acceleration of 36.6 m/s 2 and I will ignore the horizontal motion.

and I will ignore the horizontal motion. For a first approximation, I will assume air resistance is negligible.

Now I can break the motion into two parts. The first part of the motion has the capsule accelerating upwards. In the second part, the capsule is still moving up but the acceleration is in the negative y-direction (due to the gravitational force) with a value of -9.8 m/s2.

Let’s start with the first part. Doesn’t that make sense? The rockets fired at a time of 7.57 seconds and turned off at 10.97 seconds (or so I am assuming). The initial y position of the capsule was 8.84 meters (this just depends on where I put the origin of my coordinate axis). At the end of this first part, the rocket has a y-position of 219.69 meters. I can write all of this as:

Here you can see in this “Real World Problem”, you don’t always have to start with t = 0 s and y = 0 m. But what I really need is the vertical velocity at the end of this first time interval. Since I know the duration of the constant acceleration, I can use the definition of acceleration to find this velocity.

The initial y-velocity is zero – so if I put in my values for the acceleration and the time, I get a vertical velocity at the end of part one with a value of 124.4 m/s.

Now moving to part 2. I know the starting position, I know the starting velocity and I know the acceleration. I don’t know the time to get to the highest point and I don’t know the distance to the highest point (but that’s what I want to find). Since I don’t know the time, I can use the following kinematic equation (this isn’t a magic equation, you can easily derive it yourself).

Since I am looking for this capsule to go its highest point, the final velocity will be v y2 = 0 m/s and the initial velocity will be the value of v y1 from part 1. Using a vertical acceleration of -9.8 m/s2 and a starting position y 1 , I get:

So, right around 1,000 meters. Notice that in my estimate the rocket only gets the capsule about 200 meters high and then it continues to travel another 800 meters after the rockets turn off. This is because the rockets did two things. They lifted the capsule up, but they also gave it a large upward speed.

But wait! What about my assumption that the air resistance was negligible? Let’s do a quick check. A basic model for air resistance says that the magnitude of this force can be expressed as:

Here the air resistance depends on the density of air (ρ), the cross sectional area (A), the drag coefficient (C), and the velocity. The cross sectional area would be a circle (and I know the diameter). Also, I know the density of air (about 1.2 kg/m3). I am going to guess at the drag coefficient. A sphere has a value of around 0.47 so I will guess this aerodynamic capsule is about 0.3. Putting all of these values in, I get an air resistance at the end of the rocket burn phase (speed of 124.4 m/s) of 3.0 x 104 Newtons. This seems crazy high, but according to SpaceX, the Dragon capsule has a mass of 6,000 kg (weight of 5.9 x 104 N). The air resistance is less than the weight of the capsule, but it’s large enough that we should probably take that into account.

More Homework

1. Create a plot showing the vertical position of the Dragon both with and without air resistance.

Of course once you have air resistance, you pretty much have to do a numerical model. Here is a quick tutorial on using air resistance in GlowScript. If you use a drag coefficient of 0.3, you should get a plot like this:

2. Horizontal Motion. Here is a plot of the horizontal position of the Dragon as it launched. Assuming the horizontal velocity is constant after the rockets fire, how far horizontally will it travel?

3. At one point, you can see the capsule descending with open parachutes along with some trees so that you can look at the motion of the capsule. Here is the data from video analysis (I scaled it for you too). How fast was the capsule moving in both the horizontal and vertical directions?

4. Based on my estimate from the video, it takes the capsule 0.8 seconds to stop when it hits the water. Using your estimation of the speed from question 3, determine a value for the impact acceleration.

One final note. I had a student ask the the other day: “Does understanding physics make looking at the world less fun since you want to analyze everything?” My answer: Of course not. I think Richard Feynman said the same thing about a flower. If you understand how a flower works, does it make it any less beautiful? I would argue that understanding things makes them more intriguing rather than less.

I know I said “one final note”, but I have one more. I think this Dragon abort video is a great example for physics. You look at it at first glance and just think “oh, that was cool”. But as you look deeper and deeper, you find all sorts of interesting things to analyze. It’s not just a simple problem, but it is simple to first approximation.