Lesson 06

The World's Biggest Jump

Can you really free-fall faster than the speed of sound? In the fall of 2012, Felix Baumgartner jumped from a balloon in the stratosphere - in a mission sponsored by Red Bull. Here we will apply some basic physics to model his jump.

We approach this mechanics problem with Newton's F=ma, and compute the position and velocity of Felix knowing the forces applied in the problem. Begin by listing the important physical effects in the problem. Why is this not as simple as dropping a ball on the table?

It is notoriously difficult to compute the atmospheric drag of an object, but we can use a crude derivation to model its effect. In terms of energy, the work done by gravity is not only going into the kinetic energy of Felix, it is also going into the kinetic energy of the air being pushed out of the way of his falling body. If we assume the air in front Felix is quickly accelerated up to his speed, then the change in kinetic energy of the air,

$dE_{air} = \frac{1}{2} dm_{air} v^2$

is related to the amount of air mass accelerated in a given interval of time:

$dm_{air} = \rho A (v\cdot dt)$

where $\rho$ is the air density and A is the cross-sectional area of the skydiver.

Our differential equation for the velocity now looks like:

${dv\over dt} = g - {\rho A\over 2m}v^2$

where $g$ is the local acceleration due to gravity and $m$ is the mass of the falling object.

EXERCISE: Use the above equation to calculate the terminal velocity of a skydiver. You will have to estimate parameters (mass of skydiver, density of air, and cross-sectional area of the skydiver).

As a first step, solve for Felix's speed considering all the terms introduced here are constants. You will need to choose the size of your time step; too large and your solution will not be accurate, too small and you will be needlessly waiting for the computer to finish millions of calculations. While there is no steadfast rule for picking a step size, some common sense usually gives a good starting point. Nonetheless, one should always check their numerical results as a function of step size as we did in Lesson 4.

In general the step size should be chosen such that the variables in the problem do not change much (less than 1%) in a given step. If there is a natural scale size in the problem, such as the nuclear lifetime in the radioactive decay problem from Lesson 3, an appropriate choice would be some small (% ?) fraction of this scale size.

EXERCISE: Write a code that plots speed of a skydiver as a function of height and compare the terminal velocity from your numerical solution to the terminal velocity derived analytically.

Once you have verified this starting problem, add the additional physical effects you identified at the beginning of the lesson.

Assignment: Write a code that outputs the Mach number of the skydiver as a function of height above the surface of the Earth. Does Felix ever go supersonic?