Calculating Maximum Speed in Free Fall

100km accelerationFree Fall Speeds

by Robert Brand and Todd Hampson

Oddly enough, there is very little information on the web for calculating the maximum speed that a craft will fall from a specific height. It is a complex calculation requiring knowledge of the shape of a craft, the size of the craft, the amount of gravitational attraction at each height, the thickness of the atmosphere and the mass of the vehicle.

Todd Hampson has done some great work in getting the information together although he has not found a simple formula for calculating atmospheric density. He has temprarily used look-up tables and that has caused some rather “jerky” graphs. He will work on embedding a formula into the equations and removing the problematic look-up tables. None the less, this is a story of our travels and thus our problems too. Eventually it will be our triumphs too, but a bumpy chart is not a major worry to me, especially as we already know the solution. Now for the fun stuff.

Calculations, Calculations and More Calculations

Getting something “just right” the first time is near impossible and this is no different. Lots of complex data and no simple formula for air density, simply because it is not linear and non anything else. Tomorrow we will add the formula into the data and smooth out the bumps.

Today let us look at the graph that is all important, but first let’s look at an version of ThunderStruck falling from 100km. We will need to do this for Phase 2 with a different craft, but let’s look at the maths.

Todd says:
– For mass of the vehicle I used 10kg.
– For the Area of the object in direction of motion (vertically downwards I am assuming for the high speed part of the fall) I calculated the cross sectional area of the cone ie: a circle using the diameter of 600mm as per the current drawings.
– For the Drag Co-efficinet there was a URL on the VUId page that pointed to an page discussing different drag co-coefficients. For a 3D cone the Cd is calculated using a formula that needs a half-vertex angle. From your drawings (cone depth 450mm, cone diameter 600mm) half-vertex angle is 33.7 degrees.

100km release; max speed

max speed for an aircraft released from 100km – from a sounding rocket apogee of 100km

In the graph above, the first part of the flight was a little more difficult than I thought as lots of things are changing as it falls ie: gravity, air density, drag etc but I’ve got there now.

The first model I have done is the 100km drop test. I need to clean up the data below 18000m but the show is well and truly over by then anyway, but I will get it right so the graph is correct (I need to be more accurate with the air density below 18km).

This says a lot. Thanks Todd. This shows that tourist flights to space at just over 100km altitude at apogee will reach a top speed of Mach 3 on their return – that is about 1,050m/s. Then without any further intervention, they will slow to a fall of about 50m/s near the ground. This shows that the Virgin Galactic trick of feathering the craft is all about stability and not speed. There is nothing that will prevent the craft from reaching this speed since there is not enough air to interfere with the acceleration. The “chunky” graph below shows that clearly. Please assume that the peaks to the left in the deceleration part of the graph are correct.

Acceleration from 100km fall and then deceleration

Acceleration from 100km fall and then deceleration

Free Fall Speeds

From the above, you can see the acceleration is flat and continuous until the craft reaches an altitude of 60km and the acceleration starts to slow. It crosses the zero point of a stable speed at about 47km and then begins to decelerate quite rapidly until it reaches 33km altitude. At this point the deceleration slows down and at 20km altitude the deceleration is slowing in the thick air. You may notice that the maximum deceleration is 38m/s/s and since we accelerate at nearly 10m/s/s when we jump from a platform, simply put every 10m/s/s equates (rule of thumb) to 1G. This means that any craft headed straight down will experience a maximum G force of about 4G. Nothing too harsh. Slowing from orbit is very different and we will eventually cover this in future posts about re-entry.

The first thing to notice is that we will never reach Mach 3 from a release at around 45km. We will achieve over Mach 1. There are a few things that we will need to play with to reach the desired Mach 1.5 and we will cover that in a future post as we look at the graph for a drop from 45km and another from 35km.