Thursday, 31 July 2014

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WHAT IS VELOCITY

Velocity is the rate of change of the position of an object, equivalent to a specification of its speed and direction of motion, e.g. 60 km/h to the north. Velocity is an important concept in kinematics, the branch of classical mechanics which describes the motion of bodies.
Velocity is a vector physical quantity; both magnitude and direction are required to define it. The scalar absolute value (magnitude) of velocity is called "speed", a quantity that is measured in metres per second (m/s or m·s−1) in the SI (metric) system. For example, "5 metres per second" is a scalar (not a vector), whereas "5 metres per second east" is a vector.
If there is a change in speed, direction, or both, then the object has a changing velocity and is said to be undergoing an acceleration .


Equation of motion


Velocity is defined as the rate of change of position with respect to time, i.e.
\boldsymbol{v} = \frac{d\boldsymbol{x}}{d\mathit{t}}
where v is velocity and x is the displacement vector. This will give the instantaneous velocity of a particle, or object, at any particular time t. Although the concept of an instantaneous velocity might at first seem counter-intuitive, it is best considered as the velocity that the object would continue to travel at if it stopped accelerating at that moment.
Although velocity is defined as the rate of change of position, it is more common to start with an expression for an object's acceleration and from there obtain an expression for velocity, which can be done by evaluating
\int \boldsymbol{a} \ d\mathit{t}
which comes from the definition of acceleration,
\boldsymbol{a} = \frac{d\boldsymbol{v}}{d\mathit{t}}
Sometimes it is easier, or even necessary, to work with the average velocity of an object, that is to say the constant velocity, that would provide the same resultant displacement as a variable velocity, v(t), over some time period Δt. Average velocity can be calculated as
\boldsymbol{\bar{v}} = \frac{\Delta\boldsymbol{x}}{\Delta\mathit{t}}
The average velocity is always less than or equal to the average speed of an object. This can be seen by realizing that while distance is always strictly increasing, displacement can increase or decrease in magnitude as well as change direction.
In terms of a displacement-time graph, the velocity can be thought of as the gradient of the tangent line to the curve at any point, and the average velocity as the gradient of the chord line between two points with t coordinates equal to the boundaries of the time period for the average velocity.

Constant acceleration

In the special case of constant acceleration, velocity can be studied using the suvat equations. By considering a as being equal to some arbitrary constant vector, it is trivial to show that
\boldsymbol{v} = \boldsymbol{u} + \boldsymbol{a}t
with v as the velocity at time t and u as the velocity at time t=0. By combining this equation with the suvat equation x=ut+at2/2, it is possible to relate the displacement and the average velocity by
\boldsymbol{x} = \frac{(\boldsymbol{u} + \boldsymbol{v})}{2}\mathit{t} = \boldsymbol{\bar{v}}\mathit{t}.
It is also possible to derive an expression for the velocity independent of time, known as the Torricelli equation, as follows:
v^{2} = \boldsymbol{v}\cdot\boldsymbol{v} = (\boldsymbol{u}+\boldsymbol{a}t)\cdot(\boldsymbol{u}+\boldsymbol{a}t)=u^{2}+2t(\boldsymbol{a}\cdot\boldsymbol{u})+a^{2}t^{2}
(2\boldsymbol{a})\cdot\boldsymbol{x} = (2\boldsymbol{a})\cdot(\boldsymbol{u}t+\frac{1}{2}\boldsymbol{a}t^{2})=2t(\boldsymbol{a}\cdot\boldsymbol{u})+a^{2}t^{2} = v^{2} - u^{2}
\therefore v^{2} = u^{2} + 2(\boldsymbol{a}\cdot\boldsymbol{x})
where v=|v| etc...
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.

Quantities that are dependent on velocity

The kinetic energy of a moving object is dependent on its velocity by the equation
E_{\text{k}} = \tfrac{1}{2}mv^{2}
where Ek is the kinetic energy and m is the mass. Kinetic energy is a scalar quantity as it depends on the square of the velocity, however a related quantity, momentum, is a vector and defined by
\boldsymbol{p}=m\boldsymbol{v}
In special relativity, the dimensionless Lorentz Factor appears frequently, and is given by
\gamma = \frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
where γ is the Lorentz factor and c is the speed of light.


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