Velocity
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In physics, velocity is defined as the rate of change of position. It is a vector physical quantity, both speed and direction are required to define it. In the SI (metric) system, it is measured in metres per second: (m/s) or ms-1. The scalar absolute value (magnitude) of velocity is speed. 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 <math>( \Delta x)</math> during a time interval <math>( \Delta t)</math> is described by the formula:
- <math>\bar{\mathbf{v}} = \frac{\Delta \mathbf{x}}{\Delta t}</math>
The rate of change of velocity is referred to as acceleration.
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[edit] Equations of motion
The instantaneous velocity vector (v) of an object that has positions x(t) at time t and x(t+<math> {\Delta t}</math>) at time t+<math> {\Delta t}</math>, can be computed as the derivative of position:
- <math>\mathbf{v} = \lim_{\Delta t \to 0}{{\mathbf{x}(t+\Delta t)-\mathbf{x}(t)} \over \Delta t}={\mathrm{d}\mathbf{x} \over \mathrm{d}t}</math>
The equation for an object's velocity can be obtained mathematically by taking the integral of the equation for its acceleration beginning from some initial period time <math> t_0 </math> to some point in time later <math>t_n</math>.
The final velocity v of an object which starts with velocity u and then accelerates at constant acceleration a for a period of time <math>( \Delta t)</math> is:
- <math>\mathbf{v} = \mathbf{u} + \mathbf{a} \Delta t </math>
The average velocity of an object undergoing constant acceleration is <math>\begin{matrix} \frac {(\mathbf{u} + \mathbf{v})}{2} \; \end{matrix}</math>, where u is the initial velocity and v is the final velocity. To find the displacement, x, of such an accelerating object during a time interval, <math>\Delta t</math>, then:
- <math> \Delta \mathbf{x} = \frac {( \mathbf{u} + \mathbf{v} )}{2}\Delta t</math>
When only the object's initial velocity is known, the expression,
- <math> \Delta \mathbf{x} = \mathbf{u} \Delta t + \frac{1}{2}\mathbf{a} \Delta t^2,</math>
can be used.
This can be expanded to give the position at any time t in the following way:
- <math> \mathbf{x}(t) = \mathbf{x}(0) + \Delta \mathbf{x} = \mathbf{x}(0) + \mathbf{u} \Delta t + \frac{1}{2}\mathbf{a} \Delta t^2,</math>
These basic equations for final velocity and displacement can be combined to form an equation that is independent of time, also known as Torricelli's equation:
- <math>v^2 = u^2 + 2a\Delta x.\,</math>
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), <math>E_{K}</math>, of a moving object is linear with both its mass and the square of its velocity:
- <math>E_{K} = \begin{matrix} \frac{1}{2} \end{matrix} mv^2.</math>
The kinetic energy is a scalar quantity.
Escape velocity is 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 Earth is approximately 11100 m/s
[edit] Polar coordinates
In polar coordinates, a two-dimensional velocity can be decomposed into a radial velocity, defined as the component of velocity away from or toward the origin (also known as velocity made good), and transverse velocity, the component of velocity along a circle centered at the origin, and equal to the distance to the origin times the angular velocity.
Angular momentum in scalar form is the distance to the origin times the transverse speed, or equivalently, the distance squared times the angular speed, with positive quantities representing counter-clockwise direction and negative quantities representing clockwise direction (in a right-handed coordinate system).
<math>Velocity = \frac{Displacement}{Time}</math>
If forces are in the radial direction only with an inverse square dependence, as in the case of a gravitational orbit, angular momentum is constant, and transverse speed is inversely proportional to the distance, angular speed is inversely proportional to the distance squared, and the rate at which area is swept out is constant. These relations are known as Kepler's laws of planetary motion
[edit] See also
- Terminal velocity
- Hypervelocity
- Four-velocity (relativistic version of velocity for Minkowski spacetime)
- Rapidity (a version of velocity additive at relativistic speeds)
- Relative velocity
[edit] References
- Halliday, David, Robert Resnick and Jearl Walker, Fundamentals of Physics, Wiley; 7 Sub edition (June 16, 2004). ISBN 0471232319.
[edit] External links
- Speed and Velocity (The Physics Classroom)
- Introduction to Mechanisms (Carnegie Mellon University)
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