**One of my lockdown pleasures** has been watching SpaceX launches.

**I find the fact** that they are broadcast live inspiring. And the fact they will (and do) stop launches even at T-1 second shows that they do not operate on a ‘let’s hope it works’ basis. It speaks to me of confidence built on the application of measurement science and real engineering prowess.

**Aside from the thrill of the launch** and the beautiful views, one of the brilliant features of these launches is that the screen view gives lots of details about the rocket: specifically it gives time, altitude and speed.

**When coupled with a little (public) knowledge** about the rocket one can get to really understand the launch. One can ask and answer questions such as:

- What is the acceleration during launch?
- What is the rate of fuel use?
- What is Max Q?

Let me explain.

**Rocket Science#1: Looking at the data**

**To do my study** I watched the video above starting at launch, about 19 minutes 56 seconds into the video. I then repeatedly paused it – at first every second or so – and wrote down the time, altitude (km) and speed (km/h) in my notebook. Later I wrote down data for every kilometre or so in altitude, then later every 10 seconds or so.

**In all I captured around 112 readings,** and then entered them into a spreadsheet (Link). This made it easy to convert the speeds to metres per second.

**Then I plotted graphs** of the data to see how they looked: overall I was quite pleased.

**The velocity graph** clearly showed the stage separation. In fact looking in detail, one can see the Main Engine Cut Off (MECO), after which the rocket slows down for stage separation, and then the Second Engine Start (SES) after which the rocket’s second stage accelerates again.

**It is also interesting** that acceleration – the slope of the speed-versus-time graph – increases up to stage separation, then falls and then rises again.

**The first stage acceleration** increases because the thrust of the rocket is almost constant – but its mass is decreasing at an astonishing 2.5 tonnes per second as it burns its fuel!

**After stage separation,** the second stage mass is much lower, but there is only one rocket engine!

**Then I plotted a graph of altitude** versus time.

**The interesting thing about this graph** is that much of the second stage is devoted to increasing the speed of the second stage at almost constant altitude – roughly 164 km above the Earth. It’s not pushing the spacecraft higher and higher – but faster and faster.

**About 30 minutes** into the flight the second stage engine re-started, speeding up again and raising the altitude further to put the spacecraft on a trajectory towards a geostationary orbit at 35,786 km.

**Rocket Science#2: Analysing the data for acceleration**

**To estimate the acceleration **I subtracted each measurement of speed from the previous measurement of speed and then divided by the time between the two readings. This gives acceleration in units of metres per second, but I thought it would be more meaningful to plot the acceleration as a multiple of the strength of Earth’s gravitational field *g *(9.81 m/s/s).

**The data as I calculated them** had spikes in because the small time differences between speed measurements (of the order of a second) were not very accurately recorded. So I smoothed the data by averaging 5 data points together.

**The acceleration increased** as the rocket’s mass reduced reaching approximately 3.5*g* just before stage separation.

**I then wondered** if I could explain that behaviour.

**To do that I looked up the launch mass**of a Falcon 9 (Data sources at the end of the article and saw that it was 549 tonnes (549,000 kg).**I then looked up the mass**of the second stage 150 tonnes (150,000 kg).**I then assumed**that the mass of the first stage was almost entirely fuel and oxidiser and guessed that the mass would decrease uniformly from*T*= 0 to MECO at*T*= 156 seconds. This gave a burn rate of 2558 kg/s – over 2.5 tonnes per second!**I then looked up the launch thrust**from the 9 rocket engines and found it was 7,600,000 newtons (7.6 MN)**I then calculated the ‘theoretical’ acceleration**using Newton’s Second Law (*a*=*F*/*m*) at each time step – remembering to decrease the mass by 2.558 kilograms per second. And also remembering that the thrust has to exceed 1 x*g*before the rocket would leave the ground!

**The theoretical line** (**– – –**) catches the trend of the data pretty well. But one interesting feature caught my eye – a period of constant acceleration around 50 seconds into the flight.

**This is caused by the Falcon 9** throttling back its engines to reduce stresses on the rocket as it experiences maximum aerodynamic pressure – so-called Max Q – around 80 seconds into flight.

**Rocket Science#3: Maximum aerodynamic pressure**

**Rocket’s look like they do **– rocket shaped – because they have to get through Earth’s atmosphere rapidly, pushing the air in front of them as they go.

**The amount of work** needed to do that is generally proportional to the three factors:

**The cross-sectional area***A*of the rocket. Narrower rockets require less force to push through the air.**The speed of the rocket**(*squared**v*^{2}). One factor of*v*arises from the fact that travelling faster requires one to move the same amount of air out of the way faster. The second factor arises because moving air more quickly out of the way is harder due to the viscosity of the air.**The air pressure***P*. The density of the air in the atmosphere falls roughly exponentially with height, reducing by approximately 63% every 8.5 km.

**The work done by the rocket** on the air results in so-called aerodynamic stress on the rocket. These stresses – forces – are expected to vary as the product of the above three factors: *A P v*^{2}. The cross-sectional area of the rocket *A* is constant so in what follows I will just look at the variation of the product *P v*^{2}.

**As the rocket rises,** the pressure falls and the speed increases. So their product *P v, *and functions like *P v*^{2}, will naturally have a maximum value.

**The importance of the maximum **of the product *P v*^{2} (known as Max Q) as a point in flight, is that if the aerodynamic forces are not uniformly distributed, then the rocket trajectory can easily become unstable – and Max Q marks the point at which the danger of this is greatest.

**The graph below** shows the variation of pressure *P* with time during flight. The pressure is calculated using:

Where the ‘1000’ is the approximate pressure at the ground (in mbar), *h *is the altitude at a particular time, and *h*_{0} is called the *scale height of the atmosphere* and is typically 8.5 km.

**I then calculated** the product* P v*^{2}, and divided by 10 million to make it plot easily.

**This calculation** predicts that Max Q occurs about 80 seconds into flight, long after the engines throttled down, and in good agreement with SpaceX’s more sophisticated calculation.

**Summary **

**I love watching the Space X launches ** and having analysed one of them just a little bit, I feel like understand better what is going on.

**These calculations** are well within the capability of advanced school students – and there are many more questions to be addressed.

*What is the pressure at stage separation?**What is the altitude of Max Q?**The vertical velocity can be calculated by measuring the rate of change of altitude with time.**The horizontal velocity can be calculated from the speed and the vertical velocity.**How does the speed vary from one mission to another?**Why does the craft aim for a particular speed?*

**And then** there’s the satellites themselves to study!

*Good luck with your investigations!*

**Resources**

- SpaceX Launch Turksat <<<My Spreadsheet
- Overview of Falcon 9 with data on weight.
- Space X data on Falcon 9
- The Falcon 9 Users Guide from Space X (pdf) – in case you were thinking of buying a ride to space…

**And finally thanks to Jon** for pointing me towards ‘Flight Club – One-Click Rocket Science‘. This site does what I have done but with a good deal more attention to detail! Highly Recommended.