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Motion graphs and derivatives

In mechanics, the derivative of the position vs. timegraph of an object is equal to the velocity of the object. In the International System of Units, the position of the moving object is measured in meters relative to the origin, while the time is measured in seconds. Placing position on the y-axis and time on the x-axis, the slope of the curve is given by:

v = \frac{\Delta y}{\Delta x} = \frac{\Delta s}{\Delta t}.

Here s is the position of the object, and t is the time. Therefore, the slope of the curve gives the change in position (in metres) divided by the change in time (in seconds), which is the definition of the average velocity (in meters per second (\begin{matrix} \frac{m}{s} \end{matrix})) for that interval of time on the graph. If this interval is made to be infinitesimally small, such that {\Delta s} becomes {ds} and {\Delta t} becomes {dt}, the result is the instantaneous velocity at time t, or the derivative of the position with respect to time.

A similar fact also holds true for the velocity vs. time graph. The slope of a velocity vs. time graph is acceleration, this time, placing velocity on the y-axis and time on the x-axis. Again the slope of a line is change in y over change in x:

a = \frac{\Delta y}{\Delta x} = \frac{\Delta v}{\Delta t}.

Where v is the velocity, measured in \begin{matrix} \frac{m}{s} \end{matrix}, and t is the time measured in seconds. This slope therefore defines the average acceleration over the interval, and reducing the interval infinitesimally gives \begin{matrix} \frac{dv}{dt} \end{matrix}, the instantaneous acceleration at time t, or the derivative of the velocity with respect to time (or the second derivative of the position with respect to time). The units of this slope or derivative are in meters per second per second (\begin{matrix} \frac{m}{s^2} \end{matrix}, usually termed "meters per second-squared"), and so, therefore, is the acceleration.

Since the acceleration of the object is the second derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the seconds cancel out and only meters remain. \begin{matrix} \frac{m}{s} \end{matrix}s = m.)

The same multiplication rule holds true for acceleration vs. time graphs. When (\begin{matrix} \frac{m}{s^2} \end{matrix}) is multiplied by time (s), velocity is obtained. (\begin{matrix} \frac{m}{s^2} \end{matrix}s = \begin{matrix} \frac{m}{s} \end{matrix}).

Variable rates of change

The expressions given above apply only when the rate of change is constant or when only the average (mean) rate of change is required. If the velocity or positions change non-linearly over time, such as in the example shown in the figure, then differentiation provides the correct solution. Differentiation reduces the time-spans used above to be extremely small and gives a velocity or acceleration at each point on the graph rather than between a start and end point. The derivative forms of the above equations are

v = \frac{ds}{dt},
a = \frac{dv}{dt}.

Since acceleration differentiates the expression involving position, it can be rewritten as a second derivative with respect to position:

a = \frac{d^2 s}{dt^2}.

Since, for the purposes of mechanics such as this, integration is the opposite of differentiation, it is also possible to express position as a function of velocity and velocity as a function of acceleration. The process of determining the area under the curve, as described above, can give the displacement and change in velocity over particular time intervals by using definite integrals:

s(t_2)-s(t_1) = \int_{t_1}^{t_2}{v}\, dt,
v(t_2)-v(t_1) = \int_{t_1}^{t_2}{a}\, dt.


HDMI (High-Definition Multimedia Interface) is a compact audio/video interface for transmitting uncompressed digital data. It is a digital alternative to consumer analog standards, such as radio frequency (RF) coaxial cable, composite video, S-Video, SCART, component video, D-Terminal, or VGA. HDMI connects digital audio/video sources (such as set-top boxes, upconvert DVD players, HD DVD players, Blu-ray Disc players, AVCHDcamcorders, personal computers (PCs), video game consoles such as the PlayStation 3, Xbox 360, and AV receivers) to compatible digital audio devices, computer monitors, video projectors, and digital televisions.

HDMI follows the EIA/CEA-861 standards. HDMI supports, on a single cable, any uncompressed TV or PC video format, including standard, enhanced, and high-definition video; up to 8 channels of compressed or uncompressed digital audio; a Consumer Electronics Control (CEC) connection; and an Ethernet data connection. The CEC allows HDMI devices to control each other when necessary and allows the user to operate multiple devices with one remote control handset. Because HDMI is electrically compatible with the CEA-861 signals used by Digital Visual Interface (DVI), no signal conversion is necessary, nor is there a loss of video quality when a DVI-to-HDMI adapter is used. As an uncompressed CEA-861 connection, HDMI is independent of the various digital television standards used by individual devices, such as ATSC and DVB, as these are encapsulations of compressedMPEG video streams (which can be decoded and output as an uncompressed video stream on HDMI). The HDMI standard was not designed to include passing closed caption data (for example, subtitles) to the television for decoding. As such, any closed caption stream has to be decoded and included as an image in the video stream(s) prior to transmission over an HDMI cable to be viewed on the DTV. This limits the caption style (even for digital captions) to only that decoded at the source prior to HDMI transmission. This also prevents closed captions when transmission over HDMI is required for upconversion. For example, a DVD player sending an upscaled 720p/1080i format via HDMI to an HDTV has no method to pass Closed Captioning data so that the HDTV can decode as there is no line 21 VBI in that format.

HDMI products started shipping in late 2003. Over 850 consumer electronics and PC companies have adopted the HDMI specification (HDMI Adopters). In Europe, either DVI-HDCP or HDMI is included in the HD ready in-store labeling specification for TV sets for HDTV, formulated by EICTA with SES Astra in 2005. HDMI began to appear on consumerHDTVcamcorders and digital still cameras in 2006. Shipments of HDMI were expected to exceed that of DVI in 2008, driven primarily by the consumer electronics market.


The HDMI Founders are Hitachi, Matsushita Electric Industrial (Panasonic/National/Quasar), Philips, Silicon Image, Sony, Thomson (RCA), and Toshiba. Digital Content Protection, LLC provides HDCP (which was developed by Intel) for HDMI. HDMI has the support of motion picture producers Fox, mathematics, a periodic function is a function that repeats its values in regular intervals or periods. The most important examples are the trigonometric functions, which repeat over intervals of length 2π. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function which is not periodic is called aperiodic.


A function f is said to be periodic if (for some nonzero constant P) we have

f(x+P) = f(x) \,\!

for all values of x. The least positive constant P with this property is called the period. A function with period P will repeat on intervals of length P, and these intervals are sometimes also referred to as periods.

Geometrically, a periodic function can be defined as a function whose graph exhibits translational symmetry. Specifically, a function f is periodic with period P if the graph of f is invariant under translation in the x-direction by a distance of P. This definition of periodic can be extended to other geometric shapes and patterns, such as periodic tessellations of the plane.

A function that is not periodic is called aperiodic.


For example, the sine function is periodic with period 2π, since

\sin(x + 2\pi) = \sin(x) \,\!

for all values of x. This function repeats on intervals of length 2π (see the graph to the right).

Everyday examples are seen when the variable is time; for instance the hands of a clock or the phases of the moon show periodic behaviour. Periodic motion is motion in which the position(s) of the system are expressible as periodic functions, all with the same period.

For a function on the real numbers or on the integers, that means that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.

A simple example of a periodic function is the function f that gives the "fractional part" of its argument. Its period is 1. In particular,

f( 0.5 ) = f( 1.5 ) = f( 2.5 ) = ... = 0.5.

The graph of the function f is the sawtooth wave.

The trigonometric functions sine and cosine are common periodic functions, with period 2Ï€ (see the figure on the right). The subject of Fourier series investigates the idea that an 'arbitrary' periodic function is a sum of trigonometric functions with matching periods.

According to the definition above, some exotic functions, for example the Dirichlet function, are also periodic; in the case of Dirichlet function, any nonzero rational number is a period.


If a function f is periodic with period P, then for all x in the domain of f and all integers n,

f(x + nP) = f(x).

If f(x) is a function with period P, then f(ax+b), where a is a positive constant, is periodic with period P/a. For example, f(x)=sinx has period 2Ï€, therefore sin(5x) will have period 2Ï€/5.

Double-periodic functions

A function whose domain is the complex numbers can have two incommensurate periods without being constant. The elliptic functions are such functions. ("Incommensurate" in this context means not real multiples of each other.)

Complex example

Using complex variables we have the common period function:

e^{kix} = \cos{kx} + i\sin{kx}

As you can see, since the cosine and sine functions are periodic, and the complex exponential above is made up of cosine/sine waves, then the above (actually Euler's formula) has the following property. If L is the period of the function then:

L = 2\pi/k


Antiperiodic functions

One common generalization of periodic functions is that of antiperiodic functions. This is a function f such that f(x + P) = −f(x) for all x. (Thus, a P-antiperiodic function is a 2P-periodic function.)

Bloch-periodic functions

A further generalization appears in the context of Bloch waves and Floquet theory, which govern the solution of various periodic differential equations. In this context, the solution (in one dimension) is typically a function of the form:

f(x+P) = e^{ikP} f(x) \,\!

where k is a real or complex number (the Bloch wavevector or Floquet exponent). Functions of this form are sometimes called Bloch-periodic in this context. A periodic function is the special case k = 0, and an antiperiodic function is the special case k = Ï€/P.

Quotient spaces as domain

In signal processing you encounter the problem, that Fourier series represent periodic functions and that Fourier series satisfy convolution theorems (i.e. convolution of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a physics, a squeezed coherent state is any state of the quantum mechanicalHilbert space such that the uncertainty principle is saturated. That is, the product of the corresponding two operators takes on its minimum value:

\Delta x \Delta p = \frac{\hbar}2

The simplest such state is the ground state |0\rangle of the quantum harmonic oscillator. The next simple class of states that satisfies this identity are the family of coherent states |\alpha\rangle.

Often, the term squeezed state is used for any such state with \Delta x \neq \Delta p in "natural oscillator units". The idea behind this is that the circle denoting a coherent state in a quadrature diagram (see below) has been "squeezed" to an ellipse of the same area.

Mathematical definition

The most general wave function that satisfies the identity above is the squeezed coherent state (we work in units with \hbar=1)

\psi(x) = C\,\exp\left(-\frac{(x-x_0)^2}{2 w_0^2} + i p_0 x\right)

where C,x_0,w_0,p_0 are constants (a normalization constant, the center of the wavepacket, its width, and its average momentum). The new feature relative to a coherent state is the free value of the width w_0, which is the reason why the state is called "squeezed".

The squeezed state above is an eigenstate of a linear operator

\hat x + i\hat p w_0^2

and the corresponding eigenvalue equals x_0+ip_0 w_0^2. In this sense, it is a generalization of the ground state as well as the coherent state.

Examples of squeezed coherent states

Depending on at which phase the state's quantum noise is reduced, one can distinguish amplitude-squeezed and phase-squeezed states or general quadrature squeezed states. If no coherent excitation exists the state is called a squeezed vacuum. The figures below give a nice visual demonstration of the close connection between squeezed states and Heisenberg's uncertainty relation: Diminishing the quantum noise at a specific quadrature (phase) of the wave has as a direct consequence an enhancement of the noise of the complementary quadrature, that is, the field at the phase shifted by \pi/2.

From the top:

  • Vacuum state
  • Squeezed vacuum state
  • Phase-squeezed state
  • arbitrary squeezed state
  • Amplitude-squeezed state

As can be seen at once, in contrast to the coherent state the quantum noise is not independent of the phase of the light wave anymore. A characteristic broadening and narrowing of the noise during one oscillation period can be observed. The wave packet of a squeezed state is defined by the square of the wave function introduced in the last paragraph. They correspond to the probability distribution of the electric field strength of the light wave. The moving wave packets display an oscillatory motion combined with the widening and narrowing of their distribution: the "breathing" of the wave packet. For an amplitude-squeezed state, the most narrow distribution of the wave packet is reached at the field maximum, resulting in an amplitude that is defined more precisely than the one of a coherent state. For a phase-squeezed state, the most narrow distribution is reached at field zero, resulting in an average phase value that is better defined than the one of a coherent state.

In phase space, quantum mechanical uncertainties can be depicted by Wigner distribution Wigner quasi-probability distribution. The intensity of the light wave, its coherent excitation, is given by the displacement of the Wigner distribution from the origin. A change in the phase of the squeezed quadrature results in a rotation of the distribution.

Photon number distributions and phase distributions of squeezed states

The squeezing angle, that is the phase with minimum quantum noise, has a large influence on the photon number distribution of the light wave and its phase distribution as well.

For amplitude squeezed light the photon number distribution is usually narrower than the one of a coherent state of the same amplitude resulting in sub-Poissonian light, whereas its phase distribution is wider. The opposite is true for the phase-squeezed light, which displays a large intensity (photon number) noise but a narrow phase distribution. Nevertheless the statistics of amplitude squeezed light was not observed directly with photon number resolving detector due to experimental difficulty.

For the squeezed vacuum state the photon number distribution displays odd-even-oscillations. This can be explained by the mathematical form of the squeezing operator, that resembles the operator for two-photon generation and annihilation processes. Photons in a squeezed vacuum state are more likely to appear in pairs.

Experimental realizations of squeezed coherent states

There has been a whole variety of successful demonstrations of squeezed states. The most prominent ones were experiments with light fields using lasers and non-linear optics (see optical parametric oscillator). This is achieved by a simple process of four-wave mixing with a Chi-3 crystal. Squeezed states have also been realized via motional states of an ion in a trap, phonon states in From Yahoo Answers

Question:I am an A level Physics student trying to understand circular motion. If there's a force on a particle perpendicular to the velocity, then this creates an acceleration on the particle in that direction, right? So to my mind, this should mean that the 'speed in that direction' increases from zero, and the particle keeps the speed it originally had AS WELL, at 90 degrees to the new increased speed, which means an overall increase in speed due to Pythagoras' theorem... but no, the speed stays the same, and only its direction changes. Where am I going wrong here? I can do the maths and get the answers out, but I don't understand just HOW it works.

Answers:First just look at the definition of different components of circular motion: http://theory.uwinnipeg.ca/mod_tech/node45.html Now look at the following examples of uniform and non-unifrm circular motion and hope it helps: http://www.lightandmatter.com/html_books/1np/ch09/ch09.html

Question:I've read and re-read all the laws of motion and thermodynamics, blah blah blah. And yet, the earth and the universe keep spinning... Of the 4 fundamental forces of nature, gravity is not understood. Given that we DON'T understand everything about our universe, how can it be said that it can't exist when our own earth is a prime example of torque-producing perpetual motion (angular momentum or whatever)? OK, let's redefine "perpetual" as "many lifetimes" versus the strict "eternity" version. For example, without ANY energy input (think practical versus all the energy required to lift billions of gallons of water into the air and release it as rain), Hoover dam will continue to produce electricity until there's a physical defect in the structure/equipment or it stops raining. I definitely should have expanded my question to incorporate a practical application rather than a 100% strictly scientific explanation. My bad. Remember that the best scientific thinking and knowledege available 150 years ago stated that a heavier-than-air object could not fly, yet nature was replete with examples of it.

Answers:You don't understand what 'perpetual motion' means. First, it doesn't mean 'forever' - that's rather a long time. It refers to a closed system, such as a machine, that is given an intitial charge of energy to get it started, but after that continues to operate by the feedback of part of its output to its input, and thus will operate until we turn it off or something wears out. Obviousy, there will be losses due to friction, for example, so the system must amplify the energy it starts with to compensate and produce a surplus. Nobody has yet figured out a way to do that.

Question:For a science project we have to do a project on how Newton's laws and friction have to do with a sport. My friends and i chose to make our video about swimming but were not excatly sure of all the forces and things. Help is appreciated. Thanks!

Answers:You've chosen a tough example because fluid dynamics is a really messy science. A lot of the factors that determine, say, velocity through/over water have to be determined experimentally in the lab and then extrapolated to real world systems. But in general.... The max velocity V of the swimmer occurs where the drag force D = T the thrust of the swimmer. The equation for D is different depending on if the swimmer is totally immersed or only partially immersed in the water. But for both cases, drag varies as the square of the velocity v the swimmer is going through the water. So in both cases, we can write D = kv^2, where k is a swimmer and swim style specific constant. The k value will differ from swimmer to swimmer, and style to style. Drag comes from two sources: friction flow and static pressure. The friction flow is caused by the viscosity of the fluid passing over the contact area of the swimmer, in this case. That viscosity sort of pulls against the direction of the swimmer; thus the drag pulling on the swimmer. The static pressure is caused when the cross sectional area of the swimmer goes head on through the fluid. That builds up a pressure across the profile of the swimmer perpendicular to the direction of the swim. Slim swimmers have less cross sectional area; so they have less drag from that source. When the swimmer's velocity hits V, that means no more acceleration. That happens when f = ma = 0 = T - D. Which means a = 0 is the acceleration and that of course is the definition of terminal velocity, when there is no more acceleration. This is also the maximum velocity the swimmer will be able to go. f is the net force acting on the swimmer with mass m. As you can see f = ma is Newt's second law. Unfortunately, what goes into k is swimmer and style specific; so we can't come up with specific drag numbers. And finding a swimmer's thrust T would also require experimentation that is probably not doable for you. As I said, you picked a messy domain, swimming, for your video.

Question:I'm doing a project on it. If you could give a definition and example, that would be perfect!thx

Answers:First Law: Any object will remain either stationary or in motion at constant velocity unless there is an external force applied to it. Second Law: The sum of all external forces on an object is equal to the product of its mass and the acceleration experienced by the object. The second law is the basis of dynamics. The first law is a special case of the second law in which the total sum of forces is zero and, thus, the acceleration is zero which implies that an object will stay stationary or in motion at constant velocity. In other words, the first law is contained in the second law. The Third Law states that for every force applied on an object there will be an equivalent force applied back as a reaction. This law is the basis of statics and properly predicts that two objects cannot occupy the same space at the same time.

From Youtube

Simple Harmonic Motion for beginners - in 3 minutes :Introducing the idea of SHM with some examples and basic definitions.

Newton's Laws Of Motion (2) : Force, Mass And Acceleration :ESA Science - Newton In Space (Part 2): Newton's Second Law of Motion - Force, Mass And Acceleration. Newton's laws of motion are three physical laws that form the basis for classical mechanics. They have been expressed in several different ways over nearly three centuries. --- Please subscribe to Science & Reason: www.youtube.com www.youtube.com www.youtube.com --- The laws describe the relationship between the forces acting on a body and the motion of that body. They were first compiled by Sir Isaac Newton in his work "Philosophi Naturalis Principia Mathematica", first published on July 5, 1687. Newton used them to explain and investigate the motion of many physical objects and systems. For example, in the third volume of the text, Newton showed that these laws of motion, combined with his law of universal gravitation, explained Kepler's laws of planetary motion. --- Newton's Second Law of Motion: A body will accelerate with acceleration proportional to the force and inversely proportional to the mass. Observed from an inertial reference frame, the net force on a particle is equal to the time rate of change of its linear momentum: F = d(mv)/dt. Since by definition the mass of a particle is constant, this law is often stated as, "Force equals mass times acceleration (F = ma): the net force on an object is equal to the mass of the object multiplied by its acceleration." History of the second law Newton's Latin wording for the second law is: "Lex II: Mutationem ...