group velocity

Examples of group velocity in the following topics:

• Wavelength, Freqency in Relation to Speed

  • They also have two kinds of velocity: phase and group velocity.
  • This is the velocity at which the phase of any one frequency component of the wave travels.
  • Finally, the group velocity of a wave is the velocity with which the overall shape of the waves' amplitudes — known as the modulation or envelope of the wave — propagates through space.
  • This shows a wave with the group velocity and phase velocity going in different directions.
  • (The group velocity is positive and the phase velocity is negative. )

• Water Waves

  • The deep-water group velocity is half the phase velocity.
  • In shallow water for wavelengths larger than about twenty times the water depth (as often found near the coast), the group velocity is equal to the phase velocity.
  • We see a wave propagating in the direction of the phase velocity.
  • The wave can be thought to be made up of planes orthogonal to the direction of the phase velocity.

• Defining Kinematics

  • Kinematics is the study of the motion of points, objects, and groups of objects without considering the causes of its motion.
  • Kinematics is the branch of classical mechanics that describes the motion of points, objects and systems of groups of objects, without reference to the causes of motion (i.e., forces).
  • To describe motion, kinematics studies the trajectories of points, lines and other geometric objects, as well as their differential properties (such as velocity and acceleration).
  • The study of kinematics can be abstracted into purely mathematical expressions, which can be used to calculate various aspects of motion such as velocity, acceleration, displacement, time, and trajectory.
  • The physical quantities relevant to the motion of a particle include: mass m, position r, velocity v, acceleration a.

• Elastic Collisions in One Dimension

  • Consider a first particle with mass $m_{1}$ and velocity $v_{1i}$ and a second particle with mass $m_{2}$ and velocity $v_{2i}$.
  • Therefore, the velocities of particles 1 and 2 after the collision ($v_{1f}$ and $v_{2f}$ respectively) will be related to the initial velocities by:
  • In our case, we will solve for the final velocities of the two particles.
  • By grouping like terms and canceling out the ½ terms, we can rewrite our conservation of kinetic energy equation as:
  • By grouping like terms from our conservation of momentum equation we can find:

• Relative Velocity

  • Relative velocity is the velocity of an object B measured with respect to the velocity of another object A, denoted as $v_{BA}$.
  • Relative velocity is the velocity of an object B, in the rest frame of another object A.
  • Is the velocity of the fly, $u$, the actual velocity of the fly?
  • No, because what you measured was the velocity of the fly relative to the velocity of the boat.
  • The velocity that you observe the man walking in will be the same velocity that he would be walking in if you both were on land.

• Instananeous Velocity: A Graphical Interpretation

  • Instantaneous velocity is the velocity of an object at a single point in time and space as calculated by the slope of the tangent line.
  • Typically, motion is not with constant velocity nor speed.
  • However, changing velocity it is not as straightforward.
  • Since our velocity is constantly changing, we can estimate velocity in different ways.
  • Motion is often observed with changing velocity.

• Relativistic Addition of Velocities

  • A velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.
  • A velocity-addition formula is an equation that relates the velocities of moving objects in different reference frames.
  • As Galileo Galilei observed in 17th century, if a ship is moving relative to the shore at velocity $v$, and a fly is moving with velocity $u$ as measured on the ship, calculating the velocity of the fly as measured on the shore is what is meant by the addition of the velocities $v$ and $u$.
  • Since this is counter to what Galileo used to add velocities, there needs to be a new velocity addition law.
  • This change isn't noticeable at low velocities but as the velocity increases towards the speed of light it becomes important.

• Addition of Velocities

  • Relative velocities can be found by adding the velocity of the observed object to the velocity of the frame of reference it was measured in.
  • As learned in a previous atom, relative velocity is the velocity of an object as observed from a certain frame of reference.
  • demonstrates the concept of relative velocity.
  • When she throws the snowball forward at a speed of 1.5 m/s, relative to the sled, the velocity of the snowball to the observer is the sum of the velocity of the sled and the velocity of the snowball relative to the sled:
  • The magnitude of the observed velocity from the shore is the square root sum of the squared velocity of the boat and the squared velocity of the river.

• Flow Rate and Velocity

  • A fluid in motion has a velocity, just as a solid object in motion has a velocity.
  • Like the velocity of a solid, the velocity of a fluid is the rate of change of position per unit of time.
  • The flow velocity vector is a function of position, and if the velocity of the fluid is not constant then it is also a function of time.
  • In SI units, fluid flow velocity is expressed in terms of meters per seconds.
  • The magnitude of the fluid flow velocity is the fluid flow speed.

• Constant Velocity

  • An object moving with constant velocity must have a constant speed in a constant direction.
  • Motion with constant velocity is one of the simplest forms of motion.
  • You can also obtain an object's velocity if you know its trace over time.
  • In graphical terms, the velocity can be interpreted as the slope of the line.
  • Examine the terms for constant velocity and how they apply to acceleration