Velocity vector quantity

Velocity The term kinematics refers to the quantitative description of fluid motion or deformation. The rate of deformation depends on the distribution of velocity within the fluid. Fluid velocity v is a vector quantity, with three cartesian components i , and v.. The velocity vector is a function of spatial position and time. A steady flow is one in which the velocity is independent of time, while in unsteady flow v varies with time. 

Depending on the structure of the optical probe, components of vector quantities (velocity field, displacement field) and their signs can be distinguished in measurements, ensuring directional sensitivity. 

Velocity. A measure of the instantaneous rate of change of position in space with respect to time. Velocity is a vector quantity. 

Conservation of Momentum. If the mass of a body or system of bodies remains constant, then Newton s second law can be interpreted as a balance between force and the time rate of change of momentum, momentum being a vector quantity defined as the product of the velocity of a body and its mass. 

In most situations a fluid would be turbulent implying that the velocity vector, as well as the concentration c exhibits considerable variability on time scales smaller than those of prime interest. This situation can be described by writing these quantities as the sum of an average quantity (normally a time average) and a perturbation... 

Force and velocity are however both vector quantities and in applying the momentum balance equation, the balance should strictly sum all the effects in three dimensional space. This however is outside the scope of this text and the reader is referred to more standard works in fluid dynamics. 

To provide a mathematical description of a particle in space it is essential to specify not only its mass, but also its position (perhaps with respect to an arbitrary origin), as well as its velocity (and hence its momentum). Its mass is constant and thus independent of its position and velocity, at least in the absence of relativistic effects. It is also independent of the system of coordinates used to locate it in space. Its position and velocity, on the other hand, which have direction as well as magnitude, are vector quantities. Their descriptions depend on the choice of coordinate system. In this chapter Heaviside s notation will be followed, viz. a scalar quantity is represented by a symbol in plain italics, while a vector is printed in bold-face italic type. 

To sum up, the basic idea of the Doppler-selected TOF technique is to cast the differential cross-section S ajdv3 in a Cartesian coordinate, and to combine three dispersion techniques with each independently applied along one of the three Cartesian axes. As both the Doppler-shift (vz) and ion velocity (vy) measurements are essentially in the center-of-mass frame, and the (i j-componcnl, associated with the center-of-mass velocity vector can be made small and be largely compensated for by a slight shift in the location of the slit, the measured quantity in the Doppler-selected TOF approach represents directly the center-of-mass differential cross-section in terms of per velocity volume element in a Cartesian coordinate, d3a/dvxdvydvz. As such, the transformation of the raw data to the desired doubly differential cross-section becomes exceedingly simple and direct, Eq. (11). 

Here, u, v, and w are the components of the velocity vector in the x, y, and z directions, respectively. Note that velocity is treated as a vector quantity, so that the vector sum ui + vj + wk (where i, j, and k are the unit vectors in the x, y, and z directions) represents both the direction and magnitude of the fluid velocity at a particular position and time. The symbol P represents fluid pressure, p is the fluid viscosity, p is the fluid density, and the F parameters are the components of a body force acting on the fluid in the x, y, and z directions. (A body force is a force that acts on the fluid as a result of its mass rather than its surface area gravity is the most common body force.)... 

Figure 1,7b shows the corresponding process in which a jet of liquid flows into the tank. In this case, the rate of addition of mass Af is simply the mass flow rate. If the x-component of the jet s velocity is vx then the rate of flow of x-momentum into the tank is Mvx. Note that the mass flow rate Af is a scalar quantity and is therefore always positive. The momentum is a vector quantity by virtue of the fact that the velocity is a vector. 

The angular velocity vector tok and the linear velocity vk of the reference point of the rigid body k (Fig. 2.1b) are calculated recursively from the corresponding quantities of the preceding rigid body p(lc) ... 

Equation (2.2) can be considered as the fundamental governing equation for the concentration of an inert constituent in a turbulent flow. Because the flow in the atmosphere is turbulent, the velocity vector u is a random function of location and time. Consequently, the concentration c is also a random fimction of location and time. Thus, the dispersion of a pollutant (or tracer) in the atmosphere essentiaUy involves the propagation of the species molecules through a random medium. Even if the strength and spatial distribution of the source 5 are assumed to be known precisely, the concentration of tracer resulting from that source is a random quantity. The instantaneous, random concentration, c(x, y, z, t), of an inert tracer in a turbulent fluid with random velocity field u( c, y, z, t) resulting from a source distribution S x, y, z, t) is described by Eq. (2.2). 

Since velocity is a vector quantity, it is usually necessary to identify the component of the velocity, as was done for the rectangular Cartesian coordinate system in Eq. (1). The value of the integral as it differs from zero may be employed as a measure of the accuracy with which average characteristics (Kl) of the stream may be used to describe the macroscopic aspects of turbulence. Such methods do not yield results of practical significance when applied to the solution of the Navier-Stokes equations. 

Of all the macroscopic quantities in our model, the hydrodynamic density p, flow velocity vector u = (ua), and thermodynamic energy E, have the unique property of being produced by additive invariants of the microscopic motion. The latter, also called sum functions4 and summation invariants,5 occur at an early stage in most treatments. The precise formulation follows. 

Scalar quantities have magnitude only. Distance, height, mass, and age are examples of scalar quantities. Note that units typically need to be specified with scalar quantities. When indicating one s age, for example, we usually add the word years after the numerical value. Vector quantities have magnitude and direction. Once again the proper units need to be specified for vectors, and, in order for a complete understanding, the direction of the vector must also be specified. Examples of vectors include velocity and weight. 

Force, momentum, velocity and acceleration are examples of vector quantities (they have a direction and a magnitude) and are written in this book with an arrow over them. Other physical quantities (for example, mass and energy) which do not have a direction will be written without an arrow. The directional nature of vector quantities is often quite important. Two cars moving with the same velocity will never collide, but two cars with the same speed (going in different directions) certainly might ... 

The motions of two bodies connected by an attractive force can also include rotation. In the simplest case (for example, rotation of the Earth around the Sun) one of the two bodies is much more massive than the other, and the heavier body hardly moves. The attractive force causes an acceleration through Newton s Second Law, but this does not necessarily imply that the speed changes—for a perfectly circular orbit the speed is constant. Velocity is a vector quantity, and so a change in direction is an acceleration as well. 

The dynamics of the two-particle problem can be separated into center-of-mass motion and relative motion with the reduced mass /i = morn s/(rnp + me), of the two particles. The kinetic energy of the relative motion is a conserved quantity. The outcome of the elastic collision is described by the deflection angle of the trajectory, and this is the main quantity to be determined in the following. The deflection angle, X, gives the deviation from the incident straight line trajectory due to attractive and repulsive forces. Thus, x is the angle between the final and initial directions of the relative velocity vector for the two particles. 

Then (since velocity is also a vector quantity)... 

Notice that each set of terms in the equation contains one vector quantity (i.e., each term specifies a direction as well as a magnitude). Also notice that vx represents the absolute particle velocity whereas v2 is the particle velocity relative to the medium velocity. Thus if a, is the medium velocity, v2 = u -... 

The momentum of an object is the mass of the object multiplied by the velocity of the object. The mass will often be measured in kilograms (kg) and the velocity, in meters per second (m/s), so the momentum will be measured in kilogram meters per second (kg m/s). Because velocity is a vector quantity, meaning that the direction is part of the quantity, momentum is also a vector. Just like the velocity, to completely specify the momentum of an object one must also give the direction. 

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