Free fall

Commander David Scott conducting an experiment during the Apollo 15 moon landing.

In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it and moves along a geodesic. The present article only concerns itself with free fall in the Newtonian domain.

An object in the technical sense of free fall may not necessarily be falling down in the usual sense of the term. An object moving upwards would not normally be considered to be falling, but if it is subject to the force of gravity only, it is said to be in free fall. The moon is thus in free fall.

In a uniform gravitational field, in the absence of any other forces, gravitation acts on each part of the body equally and this is weightlessness, a condition that also occurs when the gravitational field is zero (such as when far away from any gravitating body). A body in free fall experiences "0 g".

The term "free fall" is often used more loosely than in the strict sense defined above. Thus, falling through an atmosphere without a deployed parachute, or lifting device, is also often referred to as free fall. The aerodynamic drag forces in such situations prevent them from producing full weightlessness, and thus a skydiver's "free fall" after reaching terminal velocity produces the sensation of the body's weight being supported on a cushion of air.

History

In the Western world prior to the 16th century, it was generally assumed that the speed of a falling body would be proportional to its weight—that is, a 10 kg object was expected to fall ten times faster than an otherwise identical 1 kg object through the same medium. The ancient Greek philosopher Aristotle (384–322 BC) discussed falling objects in what was perhaps the first book on mechanics.

The Italian scientist Galileo Galilei (1564–1642) subjected the Aristotelian theories to experimentation and careful observation. He then combined the results of these experiments with mathematical analysis in an unprecedented way.

In a tale that may be apocryphal, Galileo (or an assistant, more likely) dropped two objects of unequal mass from the Leaning Tower of Pisa. Given the speed at which such a fall would occur, it is doubtful that Galileo could have extracted much information from this experiment. Most of his observations of falling bodies were really of bodies rolling down ramps. This slowed things down enough to the point where he was able to measure the time intervals with water clocks and his own pulse (stopwatches having not yet been invented). This he repeated "a full hundred times" until he had achieved "an accuracy such that the deviation between two observations never exceeded one-tenth of a pulse beat."

Examples

Examples of objects in free fall include:

• A spacecraft (in space) with propulsion off (e.g. in a continuous orbit, or on a suborbital trajectory (ballistics) going up for some minutes, and then down).
• An object dropped at the top of a drop tube.
• An object thrown upward or a person jumping off the ground at low speed (i.e. as long as air resistance is negligible in comparison to weight).

Technically, an object is in free fall even when moving upwards or instantaneously at rest at the top of its motion. If gravity is the only influence acting, then the acceleration is always downward and has the same magnitude for all bodies, commonly denoted $g$.

Since all objects fall at the same rate in the absence of other forces, objects and people will experience weightlessness in these situations.

Examples of objects not in free fall:

• Flying in an aircraft: there is also an additional force of lift.
• Standing on the ground: the gravitational force is counteracted by the normal force from the ground.
• Descending to the Earth using a parachute, which balances the force of gravity with an aerodynamic drag force (and with some parachutes, an additional lift force).

The example of a falling skydiver who has not yet deployed a parachute is not considered free fall from a physics perspective, since he experiences a drag force that equals his weight once he has achieved terminal velocity (see below). However, the term "free fall skydiving" is commonly used to describe this case in everyday speech, and in the skydiving community. It is not clear, though, whether the more recent sport of wingsuit flying fits under the definition of free fall skydiving.

Measured fall time of a small steel sphere falling from various heights. The data is in good agreement with the predicted fall time of $\sqrt{2h/g}$, where h is the height and g is the free-fall acceleration due to gravity.

Near the surface of the Earth, an object in free fall in a vacuum will accelerate at approximately 9.8 m/s², independent of its mass. With air resistance acting on an object that has been dropped, the object will eventually reach a terminal velocity, which is around 53 m/s (195 km/h or 118 mph^[1]) for a human skydiver. The terminal velocity depends on many factors including mass, drag coefficient, and relative surface area and will only be achieved if the fall is from sufficient altitude. A typical skydiver in a spread-eagle position will reach terminal velocity after about 12 seconds, during which time he will have fallen around 450 m (1,500 ft).^[1]

Free fall was demonstrated on the moon by astronaut David Scott on August 2, 1971. He simultaneously released a hammer and a feather from the same height above the moon's surface. The hammer and the feather both fell at the same rate and hit the ground at the same time. This demonstrated Galileo's discovery that, in the absence of air resistance, all objects experience the same acceleration due to gravity. (On the Moon, the gravitational acceleration is much less than on Earth, approximately 1.6 m/s².)

Free fall in Newtonian mechanics

Uniform gravitational field without air resistance

This is the "textbook" case of the vertical motion of an object falling a small distance close to the surface of a planet. It is a good approximation in air as long as the force of gravity on the object is much greater than the force of air resistance, or equivalently the object's velocity is always much less than the terminal velocity (see below).

$v(t)=-gt+v_{0}\,$

$y(t)=-\frac{1}{2}gt^2+v_{0}t+y_0$

where

$v_{0}\,$ is the initial velocity (m/s).

$v(t)\,$ is the vertical velocity with respect to time (m/s).

$y_0\,$ is the initial altitude (m).

$y(t)\,$ is the altitude with respect to time (m).

$t\,$ is time elapsed (s).

$g\,$ is the acceleration due to gravity (9.81 m/s² near the surface of the earth).

Uniform gravitational field with air resistance

Acceleration of a small meteoroid when entering the Earth's atmosphere at different initial velocities.

This case, which applies to skydivers, parachutists or any body of mass, $m$, and cross-sectional area, $A$, with Reynolds number well above the critical Reynolds number, so that the air resistance is proportional to the square of the fall velocity, $v$, has an equation of motion

$m\frac{dv}{dt}=\frac{1}{2} \rho C_{\mathrm{D}} A v^2 - mg \, ,$

where $\rho$ is the air density and $C_{\mathrm{D}}$ is the drag coefficient, assumed to be constant although in general it will depend on the Reynolds number.

Assuming an object falling from rest and no change in air density with altitude, the solution is:

$v(t) = -v_{\infty} \tanh\left(\frac{gt}{v_\infty}\right),$

where the terminal speed is given by

$v_{\infty}=\sqrt{\frac{2mg}{\rho C_D A}} \, .$

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

$y = y_0 - \frac{v_{\infty}^2}{g} \ln \cosh\left(\frac{gt}{v_\infty}\right).$

Using the figure of 56 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. However, when the air density cannot be assumed to be constant, such as for objects or skydivers 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 meteoroids falling through the Earth's upper atmosphere. HALO jumps, including Joe Kittinger's and Felix Baumgartner's record jumps (see below), and the planned Le Grand Saut, also belong in this category.^[2]

Inverse-square law gravitational field

It can be said that two objects in space orbiting each other in the absence of other forces are in free fall around each other, e.g. that the Moon or an artificial satellite "falls around" the Earth, or a planet "falls around" the Sun. Assuming spherical objects means that the equation of motion is governed by Newton's Law of Universal Gravitation, with solutions to the gravitational two-body problem being elliptic orbits obeying Kepler's laws of planetary motion. This connection between falling objects close to the Earth and orbiting objects is best illustrated by the thought experiment, Newton's cannonball.

The motion of two objects moving radially towards each other with no angular momentum can be considered a special case of an elliptical orbit of eccentricity e = 1 (radial elliptic trajectory). This allows one to compute the free-fall time for two point objects on a radial path. The solution of this equation of motion yields time as a function of separation:

$t(y)= \sqrt{ \frac{ {y_0}^3 }{2\mu} } \left(\sqrt{\frac{y}{y_0}\left(1-\frac{y}{y_0}\right)} + \arccos{\sqrt{\frac{y}{y_0}}} \right)$

where

t is the time after the start of the fall

y is the distance between the centers of the bodies

y₀ is the initial value of y

μ = G(m₁ + m₂) is the standard gravitational parameter.

Substituting y = 0 we get the free-fall time.

The separation as a function of time is given by the inverse of the equation. The inverse is represented exactly by the analytic power series:

$y( t ) = \sum_{n=1}^{ \infty } \left[ \lim_{ r \to 0 } \left( {\frac{ x^{ n }}{ n! }} \frac{\mathrm{d}^{\,n-1}}{\mathrm{ d } r ^{\,n-1}} \left[ r^n \left( \frac{ 7 }{ 2 } ( \arcsin( \sqrt{ r } ) - \sqrt{ r - r^2 } ) \right)^{ - \frac{2}{3} n } \right] \right) \right]$

Evaluating this yields:

$y(t)=y_0 \left( x - \frac{1}{5} x^2 - \frac{3}{175}x^3 - \frac{23}{7875}x^4 - \frac{1894}{3931875}x^5 - \frac{3293}{21896875}x^6 - \frac{2418092}{62077640625}x^7 - \cdots \right) \$

where

$x = \left[\frac{3}{2} \left( \frac{\pi}{2}- t \sqrt{ \frac{2\mu}{ {y_0}^3 } } \right) \right]^{2/3}$

For details of these solutions see "From Moon-fall to solutions under inverse square laws" by Foong, S. K., in European Journal of Physics, v29, 987-1003 (2008) and "Radial motion of Two mutually attracting particles", by Mungan, C. E., in The Physics Teacher, v47, 502–507 (2009).

Free fall in general relativity

In general relativity, an object in free fall is subject to no force and is an inertial body moving along a geodesic. Far away from any sources of space time curvature, where spacetime is flat, the Newtonian theory of free fall agrees with general relativity but otherwise the two disagree. The experimental observation that all objects in free fall accelerate at the same rate, as noted by Galileo and then embodied in Newton's theory as the equality of gravitational and inertial masses, and later confirmed to high accuracy by modern forms of the Eötvös experiment, is the basis of the equivalence principle, from which basis Einstein's theory of general relativity initially took off.

Record free fall parachute jumps

Joseph Kittinger starting his record-breaking skydive in 1960. His record was broken only in 2012.

In 1914, while doing demonstrations for the U.S. Army, a parachute pioneer named Tiny Broadwick deployed her chute manually, thus becoming the first person to jump free-fall.

According to the Guinness Book of Records, Eugene Andreev (USSR) holds the official FAI record for the longest free-fall parachute jump after falling for 24,500 metres (80,400 ft) from an altitude of 25,458 metres (83,524 ft) near the city of Saratov, Russia on November 1, 1962. Although later on jumpers would ascend higher altitudes, Andreev's record was set without the use of a drogue chute during the jump and therefore remains the longest genuine free fall record.^[3]

During the late 1950s, Captain Joseph Kittinger of the United States was assigned to the Aerospace Medical Research Laboratories at Wright-Patterson AFB in Dayton, Ohio. For Project Excelsior (meaning "ever upward", a name given to the project by Colonel John Stapp), as part of research into high altitude bailout, he made a series of three parachute jumps wearing a pressurized suit, from a helium balloon with an open gondola.

The first, from 76,400 feet (23,290 m) in November 1959 was a near tragedy when an equipment malfunction caused him to lose consciousness, but the automatic parachute saved him (he went into a flat spin at a rotational velocity of 120 rpm; the g-force at his extremities was calculated to be over 22 times that of gravity, setting another record). Three weeks later he jumped again from 74,700 feet (22,770 m). For that return jump Kittinger was awarded the A. Leo Stevens parachute medal.

On August 16, 1960 he made the final jump from the Excelsior III at 102,800 feet (31,330 m). Towing a small drogue chute for stabilization, he fell for 4 minutes and 36 seconds reaching a maximum speed of 614 mph (988 km/h)^[4] before opening his parachute at 14,000 feet (4,270 m). Pressurization for his right glove malfunctioned during the ascent, and his right hand swelled to twice its normal size.^[5] He set records for highest balloon ascent, highest parachute jump, longest drogue-fall (4 min), and fastest speed by a human through the atmosphere.^[6]

The jumps were made in a "rocking-chair" position, descending on his back, rather than the usual arch familiar to skydivers, because he was wearing a 60-pound (27 kg) "kit" on his behind and his pressure suit naturally formed that shape when inflated, a shape appropriate for sitting in an airplane cockpit.

For the series of jumps, Kittinger was decorated with an oak leaf cluster to his Distinguished Flying Cross and awarded the Harmon Trophy by President Dwight Eisenhower.

In 2012, the Red Bull Stratos mission took place. On October 14, 2012, Felix Baumgartner broke the records previously set by Kittinger for the highest free fall, the highest manned helium balloon flight, and the fastest free fall; he jumped from 128,100 feet (39,045 m), reaching 833.9 mph (1342 km/h) - Mach 1.24. Kittinger was a member of the mission control and helped design the capsule and suit that Baumgartner ascended and jumped in.

On October 24, 2014, Alan Eustace broke the record previously set by Baumgartner for the highest free fall. He jumped from a height of 135,908 feet (41,425 m).^[7]

Surviving falls

The severity of injury increases with the height of a free fall, but also depends on body and surface features and the manner that the body impacts on to the surface.^[8] The chance of surviving increases if landing on a surface of high deformity, such as snow.^[8]

Overall, the height at which 50% of children die from a fall is between four and five storey heights above the ground.^[9]

JAT stewardess Vesna Vulović survived a fall of 10,000 metres (33,000 ft)^[10] on January 26, 1972 when she was aboard JAT Flight 367. The plane was brought down by explosives over Srbská Kamenice in the former Czechoslovakia (now the Czech Republic). The Serbian stewardess suffered a broken skull, three broken vertebrae (one crushed completely), and was in a coma for 27 days. In an interview, she commented that, according to the man who found her, "…I was in the middle part of the plane. I was found with my head down and my colleague on top of me. One part of my body with my leg was in the plane and my head was out of the plane. A catering trolley was pinned against my spine and kept me in the plane. The man who found me, says I was very lucky. He was in the German Army as a medic during World War Two. He knew how to treat me at the site of the accident."^[11]

In World War II there were several reports of military aircrew surviving long falls: Nick Alkemade, Alan Magee, and Ivan Chisov all fell at least 5,500 metres (18,000 ft) and survived.

It was reported that two of the victims of the Lockerbie bombing survived for a brief period after hitting the ground (with the forward nose section fuselage in freefall mode), but died from their injuries before help arrived.^[12]

Juliane Koepcke survived a long free fall resulting from the December 24, 1971, crash of LANSA Flight 508 (a LANSA Lockheed Electra OB-R-941 commercial airliner) in the Peruvian rainforest. The airplane was struck by lightning during a severe thunderstorm and exploded in mid air, disintegrating two miles (3.2 km) up. Köpcke, who was 17 years old at the time, fell to earth still strapped into her seat. The German Peruvian teenager survived the fall with only a broken collarbone, a gash to her right arm, and her right eye swollen shut.^[13]

In October 1985, 11-year-old Cindy Mosey survived a free fall of between three and five hundred feet into the sea from an Air Albatross Cessna 402B, which disintegrated in mid-flight after hitting a high voltage electricity transmission line spanning the Tory Channel in New Zealand's Marlborough Sounds. She was the sole survivor of the accident, which killed eight people including all of her family. She went on to a successful career as an international kite-surfer.^[14]

As an example of 'freefall survival' that was not as extreme as sometimes reported in the press, a skydiver from Staffordshire was said to have plunged 6,000 metres without a parachute in Russia and survived. James Boole said that he was supposed to have been given a signal by another skydiver to open his parachute, but it came two seconds too late. Boole, who was filming the other skydiver for a television documentary, landed on snow-covered rocks and suffered a broken back and rib.^[15] While he was lucky to survive, this was not a case of true freefall survival, because he was flying a wingsuit, greatly decreasing his vertical speed. This was over descending terrain with deep snow cover, and he impacted while his parachute was beginning to deploy. Over the years, other skydivers have survived accidents where the press has reported that no parachute was open, yet they were actually being slowed by a small area of tangled parachute. They might still be very lucky to survive, but an impact at 80 mph (129 km/h) is much less severe than the 120 mph (193 km/h) that might occur in normal freefall.

A falling person will reach terminal velocity after about 12 seconds, falling some 450 m (1,500 ft) in that time. That person will not then fall any faster, so it makes no difference what distance they fall if it is more than 450 m—they will still reach the ground at the same speed.^[16] The speeds reached by Kittinger, Baumgartner and Eustace were faster due to the thinner atmosphere at higher altitudes. Terminal velocity depends on air resistance, so terminal velocity increases as air resistance decreases.

See also

• Reduced-gravity aircraft
• Weightlessness
• Terminal velocity
• High-altitude military parachuting
• G-force
• Micro-g environment

References

1. "Free fall graph" (PDF). Green Harbor Publications. 2010. Retrieved 14 March 2016.
2. An analysis of his and similar jumps is given in "High altitude free fall" by Mohazzabi, P. and Shea, J. in American Journal of Physics, v64, 1242 (1996).
3. Data of the stratospheric balloon launched on 8/16/1960 For EXCELSIOR III
4. "Speed of a Skydiver (Terminal Velocity)". Hyper Text Book.
5. Higgins, Matt (May 24, 2008). "20-Year Journey for 15-Minute Fall". The New York Times. Retrieved May 2, 2010.
6. Joseph W. Kittinger - USAF Museum Gathering of Eagles
7. Markoff, John (24 October 2014). "Parachutist’s Record Fall: Over 25 Miles in 15 Minutes". The New York Times. Retrieved 25 October 2014.
8. Atanasijević, T; Nikolić, S; Djokić, V (2004). "Level of total injury severity as a possible parameter for evaluation of height in fatal falls". Srpski arhiv za celokupno lekarstvo 132 (3–4): 96–8. doi:10.2298/sarh0404096a. PMID 15307311.
9. Barlow, B.; Niemirska, M.; Gandhi, R. P.; Leblanc, W. (1983). "Ten years of experience with falls from a height in children". Journal of pediatric surgery 18 (4): 509–511. doi:10.1016/S0022-3468(83)80210-3. PMID 6620098.
10. Free Fall Research
11. Interviewed by Philip Baum, Green Light Aviation Security Training & Consultancy, in Belgrade, December 2001. "Vesna Vulovic: how to survive a bombing at 33,000 feet".
12. Cox, Matthew, and Foster, Tom. (1992) Their Darkest Day: The Tragedy of Pan Am 103, ISBN 0-8021-1382-6
13. "Survivor still haunted by 1971 air crash". CNN.com. July 2, 2009. Retrieved 2009-07-02.
14. http://planecrashinfo.com/1985/1985.htm
15. BBC News , May 2009 (May 18, 2009). "Jumper survives 6,000ft free fall". Retrieved January 4, 2010.
16. Notes and figures on free fall

External links

• Unplanned Freefall? A slightly tongue-in-cheek look at surviving free-fall without a parachute.
• Free fall accidents, mathematics of free fall - detailed research on the topic
• Freefall formula calculator
• Chuteless jump survivors
• "Joseph W. Kittinger and the Highest Step in the World". Greg Kennedy. March 17, 2010. detailed account of origins and development of EXCELSIOR project
• The Way Things Fall an educational website
• H.S. Free fall lesson with videos and interactive flash animations.