This mass is based on the volume of the falling object, which given Felix’s relative size and the extremely low density of the atmosphere, is likely to be small in any case. It typically adds to the effective mass of an object in the equation. The added mass effect accounts for the additional force required to move the surrounding fluid, in this case air, around a moving object. We will ignore the added mass effect in our example. Where m is the mass of Felix and his gear. This isn’t surprising, since relative to the radius of the Earth ( 3,956.6 miles), 24.26 miles is a short distance. Now the acceleration due to gravity doesn’t vary very much over the range the jump: 9.69 meters per second when he jumps to 9.831 meters per second when he lands. The other major force working on our jumper is gravity g. Where v is velocity, C d is the drag coefficient, A is maximal cross-sectional area, and ρ( z) is air density at altitude z. Air resistance is dominated by the quadratic term for high velocities and takes the form: So how easy is it to break the sound barrier by falling? Wolfram|Alpha can’t yet easily tell us, but with a bit of help from Mathematica, we can find out.įirst we need to examine the equations of motion. We can see that there is a nice minimum for the speed of sound in the middle of Felix’s fall, ideal if one wants to break the sound barrier. Taking the interpolation, we can get a closer look at the relevant portions of Felix’s jump. Here we just extract the plot points, adjusting for the logarithm taken for our x coordinates and flipping the x and y axes. We can zoom in on the area of interest with some short Mathematica code. ![]() This is not a linear relationship, as the temperature and chemical makeup of the atmosphere varies significantly with altitude, and also affects the speed of sound: At high altitudes, because of the cold and the low density of the atmosphere, it takes longer for sound to travel. The fact that the speed of sound varies with altitude also helped. Of course, this would require him to wear an oxygenated suit to allow him to breathe, in addition to keeping him warm. ![]() With less air around him, there would be less drag, and thus he could reach a higher maximum speed. This information was important to Felix’s goal to break the sound barrier in free fall because the rate of drag is directly related to air pressure. Given this knowledge, we know that 99.67% of the world’s atmosphere lay beneath him. To put it another way, the mass of the air above 39 kilometers is only 0.32851% of the total air mass. At this layer of the atmosphere, called the stratosphere, the air pressure is only 3.3 millibars, equivalent to 0.33% of the air pressure at sea level. As Wolfram|Alpha shows us, it rises and falls depending on factors such as the decreased density of air with rising altitude, but also the absorption of UV light by the ozone layer.Īt 39 kilometers, the horizon is roughly 439 miles away. The temperature, unlike air pressure, does not change linearly with altitude at such heights. Wolfram|Alpha tells us the jump was equivalent to a fall from 4.4 Mount Everests stacked on top of each other, or falling 93% of the length of a marathon.Īt 24.26 miles above the Earth, the atmosphere is very thin and cold, only about -14 degrees Fahrenheit on average. Earlier this month, on a nice day, Felix Baumgartner jumped from 39,045 meters, or 24.26 miles, above the Earth from a capsule lifted by a 334-foot-tall helium filled balloon (twice the height of Nelson’s column and 2.5 times the diameter of the Hindenberg).
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