Editing Free fall (section)

=== Uniform gravitational field with air resistance ===
[[File:MeteorAccGraph.jpg|thumb|Acceleration of a small meteoroid when entering the Earth's atmosphere 80 km high (above which the [[Kennelly–Heaviside layer]]) at different initial velocities of 35, 25 and 15 km/s. Air pressure and air density are height-dependent.]]

This case, which applies to 1. skydivers, parachutists or any body of mass, <math>m</math>, and cross-sectional area, <math>A</math>, 2. with [[Reynolds number]] Re well above the critical Reynolds number, so that the air resistance is proportional to the square of the fall velocity, <math>v</math>, 

has an equation of vertical motion in Newton's regime
:<math>m\frac{\mathrm{d}v}{\mathrm{d}t}=mg - \frac{1}{2} \rho C_{\mathrm{D}} A v^2 \, ,</math>
where <math>\rho</math> is the [[Density of air|air density]] and <math>C_{\mathrm{D}}</math> is the [[drag coefficient]], assumed to be constant (Re > 1000) although in general it will depend on the Reynolds number.

Assuming an object falling from rest and no change in air density with altitude (ideal gas? <math display="inline">1/ \rho=RT/p</math>), the solution is:
: <math>v(t) = v_{\infty}\tanh\left(\frac{gt}{v_{\infty}}\right),</math>

where the [[terminal speed]] is given by
:<math>v_{\infty}=\sqrt{\frac{2mg}{\rho C_D A}} \, .</math>

The object's speed versus time can be integrated over time to find the vertical position as a function of time:

:<math>y = y_0 - \frac{v_{\infty}^2}{g} \ln \cosh\left(\frac{gt}{v_\infty}\right).</math>

Using the figure of 56&nbsp;m/s for the terminal velocity of a human, one finds that after 10 seconds he will have fallen 348 metres and attained 94% of terminal velocity, and after 12 seconds he will have fallen 455 metres and will have attained 97% of terminal velocity. Gravity field is (vertical) position-dependent g(y): when <math display="inline">y_0 \ll R</math>, <math display="inline">g/g_o=1-2y_0/R</math>. Linear decrease with height, small height compared to Earth's radius R = 6379 km.

However, when the air density cannot be assumed to be constant, such as for objects falling from high altitude, the equation of motion becomes much more difficult to solve analytically and a numerical simulation of the motion is usually necessary. The figure shows the forces acting on small meteoroids falling through the Earth's upper atmosphere (an acceleration of 0.1 km/s² is 10 g<sub>0</sub>). [[High-altitude military parachuting#High Altitude Low Opening – HALO|HALO jump]]s, including [[Joe Kittinger]]'s and [[Felix Baumgartner]]'s record jumps, also belong in this category.<ref>An analysis of such jumps is given in {{cite journal|title=High altitude free fall|journal= American Journal of Physics |volume= 64 |issue= 10 |page= 1242 |author1=Mohazzabi, P. |author2=Shea, J. |doi=10.1119/1.18386|bibcode=1996AmJPh..64.1242M|url=http://www.jasoncantarella.com/downloads/AJP001242.pdf|year= 1996 }}</ref>