Free fall

Free fall describes any motion of a body where gravity is the only or dominant force acting upon it, at least initially. Since this definition does not specify velocity, it also applies to objects initially moving upward. Although strictly the definition excludes motion of an object subjected to other forces such as aerodynamic drag, in nontechnical usage falling through an atmosphere without a deployed parachute or lifting device is also referred to as free fall.

Contents

Examples

Examples of objects in free fall include:

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:

The example of a falling skydiver who has not yet deployed a parachute is not considered free fall from a physics perspective, since they experience a drag force which equals their weight once they have 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.

On Earth and on the Moon

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 acceleration of gravity.

Near the surface of the Earth, an object in free fall in a vacuum will accelerate at approximately 9.8 m/s^{2}, independent of its mass. With air resistance acting upon an object that has been dropped, the object will eventually reach a terminal velocity, around 56 m/s (200 km/h or 120 mph) for a human body. 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.

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/s^{2}).

Free fall in Newtonian mechanics

Main article: 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).

Free-fall
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 bodies with Reynolds number well above the critical Reynolds number, has an equation of motion:

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

where

m is the mass of the object,
g is the gravitational acceleration (assumed constant),
CD is the drag coefficient,
A is the cross-sectional area of the object, perpendicular to air flow,
v is the fall (vertical) velocity, and
ρ is the air density.

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 velocity 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).

When the air density cannot be assume 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 Col. Joe Kittinger's record jump (see below) and the planned Le Grand Saut also belong in this category. 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, 1242-1246 (1996).

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{ 3 }{ 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) \ | \ 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

Main article: General Relativity

The experimental observation that all objects in free fall accelerate at the same rate, as noted by Galileo and confirmed to high accuracy by modern forms of the Eötvös experiment, is the basis of the Equivalence Principle, on which Einstein's theory of general relativity relies. An alternative statement of this law, as can be seen from Newton's 2nd law applied to free fall above, is that the gravitational and the inertial mass of any object are the same.

Record free fall parachute jumps

Joseph Kittinger starting his record-breaking skydive.

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 80,380 ft (24,500 m) from an altitude of 83,523 ft (25,460 m) near the city of Saratov, Russia on November 1, 1962. Though later jumpers would ascend higher, Andreev's record was set without the use of a drogue chute during the jump.[1]

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 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) [1] 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.[2] He set records for highest balloon ascent, highest parachute jump, longest drogue-fall (4 min), and fastest speed by a human through the atmosphere[3].

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-lb "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 D.F.C. and awarded the Harmon Trophy by President Dwight Eisenhower.

Surviving falls

JAT stewardess Vesna Vulović survived a fall of 33,000 feet (10,000 m)[4] 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 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." [5]

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.

Freefall is not to be confused with individuals who survive instances of various degrees of controlled flight into terrain. Such impact forces affecting these instances of survival differ from the forces which are characterized by free fall.

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.[6]

A skydiver from Staffordshire plunged 6,000 ft without a parachute in Russia and survived to tell the tale. James Boole, from Tamworth, said he was supposed to have been given a signal by another skydiver to open his parachute, but it came two seconds too late. Mr Boole, who was filming the other skydiver for a television documentary, landed on snow-covered rocks and suffered a broken back and rib.[7]

See also

References

  1. ↑ Data of the stratospheric balloon launched on 8/16/1960 For EXCELSIOR III
  2. ↑ Higgins, Matt (May 24, 2008). "20-Year Journey for 15-Minute Fall". The New York Times. http://www.nytimes.com/2008/05/24/sports/othersports/24jump.html?em&ex=1211774400&en=841aa50b9281518a&ei=5087%0A. Retrieved May 2, 2010. 
  3. ↑ Joseph W. Kittinger - USAF Museum Gathering of Eagles
  4. ↑ Free Fall Research
  5. ↑ 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". http://www.avsec.com/interviews/vesna-vulovic.htm. 
  6. ↑ Cox, Matthew, and Foster, Tom. (1992) Their Darkest Day: The Tragedy of Pan Am 103, ISBN 0-8021-1382-6
  7. ↑ BBC News , May 2009 (May 18, 2009). "Jumper survives 6,000ft free fall". http://news.bbc.co.uk/2/hi/uk_news/england/staffordshire/8056599.stm. Retrieved January 4, 2010. 

External links