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# describe the motion of freely falling body

From Wikipedia

Free fall

right|thumb|Scott conducting an experiment during the [[Apollo 15]] moon landing.

Free fall is any motion of a body where gravity is the only or dominant force acting upon it, at least initially. These conditions produce an inertial trajectory so long as gravity remains the only force. Since this definition does not specify velocity, it also applies to objects initially moving upward. Since free fall in the absence of forces other than gravity produces weightlessness or "zero-g," sometimes any condition of weightlessness due to inertial motion is referred to as free-fall. This may also apply to weightlessness produced because the body is far from a gravitating body.

Although strict technical application of 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 often referred to as free fall. The drag forces in such situations prevent them from producing full weightlessness, and thus a skydiver's "free fall" after reachingterminal velocity produces the sensation of the body's weight being supported on a cushion of air.

## Examples

thumb|A video showing objects free-falling 215 feet (65 m) down a metal well, a type of drop tube 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 going up for some minutes, and then down).
• An object dropped at the top of a drop tube.
• An object thrown upwards or a person jumping off the ground at low speed (i.e. as long as air resistance is negligible in comparison to weight). Technically, the object or person is in free fall even when moving upwards or instantaneously at rest at the top of their motion, since the acceleration is still g downwards. However in common usage "free fall" is understood to mean downwards motion.

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 acceleration 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 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

Near the surface of the Earth, an object in free fall in a vacuum will accelerate at approximately 9.8&nbsp;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&nbsp;m/s (200&nbsp;km/h or 120&nbsp;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&nbsp;m/s^{2}).

## 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

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
Ï� 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 assumed to be constant, such as for objects or skydivers falling f

Rigid body

In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it. Even though such an object cannot physically exist due to relativity, objects can normally be assumed to be perfectly rigid if they are not moving near the speed of light.

In classical mechanics a rigid body is usually considered as a continuous mass distribution, while in quantum mechanics a rigid body is usually thought of as a collection of point masses. For instance, in quantum mechanics molecules (consisting of the point masses: electrons and nuclei) are often seen as rigid bodies (see classification of molecules as rigid rotors).

## Kinematics

### Linear and angular position

The position of a rigid body is the position of all the particles of which it is composed. To simplify the description of this position, we exploit the property that the body is rigid, namely that all its particles maintain the same distance relative to each other. If the body is rigid, it is sufficient to describe the position of at least three non-collinear particles. This makes it possible to reconstruct the position of all the other particles, provided that their time-invariant position relative to the three selected particles is known. However, typically a different and mathematically more convenient approach is used. The position of the whole body is represented by:

1. the linear position or position of the body, namely the position of one of the particles of the body, specifically chosen as a reference point (for instance its center of mass or its centroid, or the origin of a coordinate system fixed to the body), together with
2. the angular position (or orientation) of the body.

Thus, the position of a rigid body has two components: linear and angular, respectively. The same is true for other kinematic and kinetic quantities describing the motion of a rigid body, such as velocity, acceleration, momentum, impulse, and kinetic energy.

The linear position can be represented by a vector with its tail at an arbitrary reference point in space (often the origin of a chosen coordinate system) and its tip at a point of interest on the rigid body (often its center of mass or centroid).

There are several ways to numerically describe the orientation of a rigid body, including a set of three Euler angles, a quaternion, or a direction cosine matrix (also referred to as a rotation matrix).

In general, when a rigid body moves, both its position and orientation vary with time. In the kinematic sense, these changes are referred to as translationandrotation, respectively. Indeed, the position of a rigid body can be viewed as a hypothetic translation and rotation (roto-translation) of the body starting from a hypothetic reference position (not necessarily coinciding with a position actually taken by the body during its motion).

### Linear and angular velocity

Velocity (also called linear velocity) and angular velocity are measured with respect to a frame of reference.

The linear velocityof a rigid body is avector quantity, equal to the time rate of change of its linear position. Thus, it is the velocity of a reference point fixed to the body. During purely translational motion (motion with no rotation), all points on a rigid body move with the same velocity. However, when motion involves rotation, the instantaneous velocity of any two points on the body will generally not be the same. Two points of a rotating body will have the same instantaneous velocity only if they happen to lay on an axis parallel to the instantaneous axis of rotation.

Angular velocityis avector quantity that describes the angular speed at which the orientation of the rigid body is changing and the instantaneous axis about which

Linear motion

Linear motion is motion along a straight line, and can therefore be described mathematically using only one spatial dimension. It can be uniform, that is, with constant velocity (zero acceleration), or non-uniform, that is, with a variable velocity (non-zero acceleration). The motion of a particle (a point-like object) along the line can be described by its position x, which varies with t (time). Linear motion is sometimes called rectilinear motion.

An example of linear motion is that of a ball thrown straight up and falling back straight down.

The average velocity v during a finite time span of a particle undergoing linear motion is equal to

v = \frac {\Delta d}{\Delta t}.

The instantaneous velocity of a particle in linear motion may be found by differentiating the position x with respect to the time variable t. The acceleration may be found by differentiating the velocity. By the fundamental theorem of calculus the converse is also true: to find the velocity when given the acceleration, simply integrate the acceleration with respect to time; to find displacement, simply integrate the velocity with respect to time.

This can be demonstrated graphically. The gradient of a line on the displacement time graph represents the velocity. The gradient of the velocity time graph gives the acceleration while the area under the velocity time graph gives the displacement. The area under an acceleration time graph gives the velocity.

Linear motion is the most basic of all motions. According to Newton's first law of motion, objects not subjected to forces will continue to move uniformly in a straight line indefinitely. Under every-day circumstances, external forces such as gravity and friction will cause objects to deviate from linear motion and can cause them to come to a rest.

For linear motion embedded in a higher-dimensional space, the velocity and acceleration should be described as vectors, made up of two parts: magnitude and direction. The direction part of these vectors is the same and is constant for linear motion, and only for linear motion

Question:Q 1.a body moving with constant speed has uniform motion while a freely falling object has non uniform motion. why? Q 2.can a body have acceleration while at rest? give eg. Q 3. what type of motion is associated with a pendulum? Q 4. what does the rate of change of angular velocity refer to? Q 5. a boy after going around a circular track of radius 'r' comes back to the starting point. a)what is the displacement of the body? b)what is the total distance covered by him?

Answers:Q1. Uniform motion definition (my guess): Same distance covered every second, which also means a constant velocity. Free fall has an increasing velocity every second. Q2. Yes. You are accelerating right now while reading this. Gravity is causing you to ALWAYS accelerate while on earth. Q3. Simple harmonic motion Q4. Angular acceleration Q5. The displacement is 0 (starting position minus ending position).The distance covered is 2*pi*r (circumference)

Question:Determined to test the law of gravity himself, a student walks off a skyscraper 180 m high, stopwatch in hand, and starts his free fall(0 initial velocity). Five sec. later, Superman arrives at the scene and dives off the roof to save the student. a.) Superman leaves the roof with an initial speed v0 that he produces by pushing himself downward from the edge of the roof with his legs of steel. He then falls with the same acceleration as any freely falling body. What must the value of v0 be so that Superman catches the student just before they reach the ground? b.)if the height of the skyscraper is less than some minimum value, even Superman can t reach the student before he hits the ground. What is this minimum height?

Answers:Working formula is S = Vo(T) + (1/2)gT^2 where S = distance travelled Vo = initial velocity T = time interval g = acceleration due to gravity = 9.8 m/sec^2 For the student, S = 0 + (1/2)(9.8)(T^2) S = 4.9T^2 For Superman, S = Vo(T - 5) + (1/2)(9.8)(T - 5)^2 For Superman to catch the student, 4.9T^2 = Vo(T - 5) + (1/2)(9.8)(T - 5)^2 4.9T^2 = Vo(T - 5) + 4.9(T^2 - 10T + 25) 4.9T^2 = Vo(T - 5) + 4.9T^2 - 49T + 122.5 Solving for Vo, Vo = (49T - 122.5)/(T - 5) -- call this Equation 3 Going back to Equation 1, since "Superman catches the student just before they reach the ground?" , then 180 = 4.9(T^2) and solving for "T" T = 6.06 sec. Substituting T = 6.06 into Equation 3, Vo = (49*6.06 - 122.5)/(6.06 - 5) Vo = 168 m/sec. << if the height of the skyscraper is less than some minimum value, even Superman can t reach the student before he hits the ground. What is this minimum height? >> The calculated values for the above were based on the given data of the problem. ASSUMING all the conditions remain the same, then if the height of the skyscraper were less than 180 m, then Superman cannot save the student before he hits the ground. Hope this helps.

Question:In the Galapagos Islands at the equator, the acceleration of a freely falling body is 9.780m/s^2, while at he latitude of Oslo, Norway, it is 9.831m/s^2. why does the acceleration differ?

Answers:Height from the Earth. The gravitational pull is greater the closer you are to the center of the Earth. Also, gravitational pull at the equator is going to be less because the Earth actually bows out at the center.

James Horner - Suite from the Motion Picture Score LEGENDS OF THE FALL (1994) :Legends of the Fall is a 1994 drama film based on the 1979 novella of the same title by Jim Harrison. It was directed by Edward Zwick and stars Brad Pitt, Anthony Hopkins and Aidan Quinn. The film won the Academy Award for Best Cinematography. The movie's timeframe spans the decade before World War I through the Prohibition era, and into the 1930s, ending with a brief scene set in 1963. The film centers on the Ludlow family of Montana, including veteran of the Indian Wars, Colonel Ludlow, his sons, Alfred, Tristan, and Samuel, and object of the brothers' love, Susannah. This movie was shot in Alberta and British Columbia, Canada. The film opened in limited release on December 23, 1994 and made $14 million in its first weekend in wide release a month later. It went on to have a final box office total of$66 million. Although released in the hopes of being an Academy Award frontrunner, the film was nominated for just three awards, in none of the major categories. It won one award, for best cinematographer John Toll. The film has a 70% positive review from critics on Rotten Tomatoes, with acclaimed critic Roger Ebert describing it as "pretty good ... with full-blooded performances and heartfelt melodrama." In one scene (at 25 min, 24 sec), the book "Report of a Reconnaissance of the Black Hills of Dakota" is shown and referred to as the work of the fictional William Ludlow. This is a real book, written by the real William Ludlow, who served as a Colonel during the Plains ...