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Derivative

In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position

Physical quantity - Wikipedia, the free encyclopedia

The seven base quantities of the International System of Quantities (ISQ) and their ... For example, the physical quantity velocity is derived from base ...

Velocity

In physics, velocity is the measurement of the rate and direction of change in position of an object. It is a vectorphysical quantity; both magnitude and direction are required to define it. The scalarabsolute value (magnitude) of velocity is speed, a quantity that is measured in meters per second (m/s or msâˆ’1) when using the SI (metric) system.

For example, "5 meters per second" is a scalar and not a vector, whereas "5 meters per second east" is a vector. The average velocity v of an object moving through a displacement ( \Delta \mathbf{x}) during a time interval ( \Delta t) is described by the formula:

\mathbf{\bar{v}} = \frac{\Delta \mathbf{x}}{\Delta t}.

The rate of change of velocity is accelerationâ€“ how an object's speed or direction changes over time, and how it is changing at a particular point in time.

## Equation of motion

The velocity vector v of an object that has positions x(t) at time t and x(t + \Delta t) at time t + \Delta t, can be computed as the derivative of position:

\mathbf{v} = \lim_{\Delta t \to 0} \over \Delta t}={\mathrm{d}\mathbf{x} \over \mathrm{d}t}.

Average velocity magnitude is always smaller than or equal to average speed of a given particle. Instantaneous velocity is always tangential to trajectory. Slope of tangent of position or displacement time graph is instantaneous velocity and its slope of chord is average velocity.

The equation for an object's velocity can be obtained mathematically by evaluating the integral of the equation for its acceleration beginning from some initial period time t_0 to some point in time later t_n.

The final velocity v of an object which starts with velocity u and then accelerates at constant acceleration a for a period of time \Delta t is:

\mathbf{v} = \mathbf{u} + \mathbf{a} \Delta t.

The average velocity of an object undergoing constant acceleration is \tfrac {(\mathbf{u} + \mathbf{v})}{2}, where u is the initial velocity and v is the final velocity. To find the position, x, of such an accelerating object during a time interval, \Delta t, then:

\Delta \mathbf{x} = \frac {( \mathbf{u} + \mathbf{v} )}{2}\Delta t.

When only the object's initial velocity is known, the expression,

\Delta \mathbf{x} = \mathbf{u} \Delta t + \frac{1}{2}\mathbf{a} \Delta t^2,

can be used.

This can be expanded to give the position at any time t in the following way:

\mathbf{x}(t) = \mathbf{x}(0) + \Delta \mathbf{x} = \mathbf{x}(0) + \mathbf{u} \Delta t + \frac{1}{2}\mathbf{a} \Delta t^2,

These basic equations for final velocity and position can be combined to form an equation that is independent of time, also known as Torricelli's equation:

v^2 = u^2 + 2a\Delta x.\,

The above equations are valid for both Newtonian mechanics and special relativity. Where Newtonian mechanics and special relativity differ is in how different observers would describe the same situation. In particular, in Newtonian mechanics, all observers agree on the value of t and the transformation rules for position create a situation in which all non-accelerating observers would describe the acceleration of an object with the same values. Neither is true for special relativity. In other words only relative velocity can be calculated.

In Newtonian mechanics, the kinetic energy (energy of motion), E_K, of a moving object is linear with both its mass and the square of its velocity:

E_{K} = \begin{matrix} \frac{1}{2} \end{matrix} mv^2.

The kinetic energy is a scalar quantity.

Escape velocityis the minimum velocity a body must have in order to escape from the gravitational field of the earth. To escape from the Earth's gravitational field an object must have greater kinetic energy than its gravitational potential energy. The value of the escape velocity from the Earth's surface is approximately 11100 m/s.

## Relative velocity

Relative velocity is a measurement of velocity between two objects as determined in a single coordinate system. Relative velocity is fundamental in both classical and modern physics, since many systems in physics deal with the relative motion of two or more particles. In Newtonian mechanics, the relative velocity is independent of the chosen inertial reference frame. This is not the case anymore with special relativity in which velocities depend on the choice of reference frame.

If an object A is moving with velocity vectorv and an object B with velocity vector w, then the velocity of object A relative to object B is defined as the difference of the two velocity vectors:

\mathbf{v}_{A\text{ relative to }B} = \mathbf{v} - \mathbf{w}

Similarly the relative velocity of object B moving with velocity w, relative to object A moving with velocity v is:

\mathbf{v}_{B\text{ relative to }A} = \mathbf{w} - \mathbf{v}

Usually the inertial frame is chosen in which the latter of the two mentioned objects is in rest.

### Scalar velocities

In the one dimensional case, the velocities are scalars and the equation is either:

\, v_{rel} = v - (-w), if the two objects are moving in opposite directions, or:
\, v_{rel} = v -(+w), if the two objects are moving in the same direction.

## Polar coordinates

In polar coordinates, a two-dimensional velocity is described by a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and an

Question:I can easily calculate derivatives but I simply can't understand what does it mean in the first place. Well, let's take this example with physics task: v(t) = 3t + 2t^2 a(t) = 3 + 4t And if we take t(time), let's say 0 and put in both velocity and acceleration functions, we get that acceleration(derivative of velocity) is equal to 3, but speed is equal to 0. So I just can't understand how exactly speed can be 0 if at the same point of time acceleration is 3?! If anyone can explain this that even a retard would understand, I would be really pleased.

Answers:Fortunately, you seem to know what you don't know - and understanding what a derivative MEANS is critical - and can be a bit confusing at first. Don't worry; you'll get it eventually. You're right that in your equation, v(0) = 0 and a(0) = 3. This means that the particle is not moving at that INSTANT t=0, but is accelerating in the positive direction. Let's change the example so that v(0) = 0 and a(0) = -9.8. (If you've dabbled in physics, that number -9.8 should be familiar to you.) What this represents is that at t=0, you've thrown a ball up in the air at sometime t < 0, and the ball hit its peak at t=0, and now the ball will be moving down at t >0. At the PEAK of your ball's motion, your ball is literally not moving at all. That's why v=0. If you were to take two photographs very, very, very close to the time when t=0, you'd see that the ball is basically in the same place, at the peak. However, the ball is still under the force of gravity, which causes an acceleration. It's not changing its position, so its velocity is zero. However, the velocity is changing, and the acceleration is -9.8 m/s^2.

Question:I'm going into junior year in high school and I'm just beginning to teach myself calculus soo this might be a stupid question but whatever. When you have a infinite as a hyperreal number in an equation and you are to find the standard parts, how do you eliminate the infinite...or do you have to? For example: When H is infinite find the standard parts of: (H+1+ )/(2H-1+3 ) How do you find the standard parts, please explain in detail since Im teaching myself! Thank you verry much. Can you just distribute H at a negative power to numerator and denominator and go on, if so, then what?

Answers:It's important to remember that we can never actually use in any calculation because is not a number but a concept. It is what we say for a quantity that grows without bounds. I think what you are attempting to do here is evaluate limits. That is, the limit of the given expression as H approaches . To do this, yes, we first divide through by H: lim{H-> } (1 + 1/H + E/H) / (2 - 1/H + 3E/H) Now we note that 1/H and E/H both approach 0 as H -> , so: lim{H-> } (1 + 0 + 0) / (2 - 0 + 3*0) 1/2

Question:ok.. i have to make a database driven website and i will be using ASP to do it. I did this before.. like 7 years ago and then never did it again. So now i don't remember most of the things i learned at that time. I have to make basic calculations of numbers from a table in an access database. Lets say i have a column called "quantity" with many rows with numbers in them. How do i add them up to print the total in a webpage. Can you give me some examples of basic functions with sum and multiplication to refresh my memory. Thanks a lot. BTW.. i don't have time to start learning PHP or other languajes.. so do not suggest anything like that.

Answers:Here is a more detailed ASP example: <% Dim conn, rs 'Assumes: connection established (using conn as variable name) qry = [your query here, enclosed in quotes] Set rs = conn.Execute(qry) While Not rs.EOF Sum = Sum + rs("quantity") Wend . . . %> HTML:

Question:1. Differentiate thet SI system of measurement from the English system of measurement. 2. For the SI system, discuss the 7 basic quantities, as well as the derived quantities. Give examples. 3. Give examples of units used in the English System. 4. What is Dimensional analysis? Briefly explain. Dont copy past from wiki please