Momentum Velocity multiplied

Now each such particle adds its change in momentum, as given above, to the total change of momentum of the gas in time t. The total change in momentum of the gas is obtained by multiplying Af by the change in momentum per particle and integratmg over all allowed values of tlie velocity vector, namely, those for which V n< 0. That is... 

Momentum Momentum is the force of motion of a moving body. Quantitatively, it is the product of its mass of the object multiplied by its velocity, p = mv. 

In contrast to the case of the homogeneous model, the accelerative term cannot be put in a simpler form because the phase velocities differ. It is therefore necessary to carry out the differentiation in the accelerative term. When this is done and the frictional component of the pressure gradient is represented using the wholly liquid two-phase multiplier, the resulting form of the momentum equation is... 

For a fuller explanation of this usage, see Ref. 5, p. 39. The expression for F in Eq. (18), when multiplied by m3 to convert it from a density in momentum space to one in velocity space, and when a2 is replaced by mkT, coincides in form with the Maxwell velocity distribution. It is,... 

Notice that these equations explicitly include derivatives and the complex conjugate f of the wavefunction. The expression for the momentum even includes i = f—W Complex numbers are not just a mathematical convenience in quantum mechanics they are central to the treatment. Equation 6.6 illustrates this point directly. Any measurement of the momentum (for example, by measuring velocity and mass) will of course always give a real number. But if the wavefunction is purely real, the integral on the right-hand side of Equation 6.6 is a real number, so the momentum is a real number multiplied by ih. The only way that product can be real is if the integral vanishes. Thus any real wavefunction corresponds to motion with no net momentum. Any particle with net momentum must have a complex wavefunction. 

The momentum of a body is simply the product of its weight or mass and its velocity. When a force acts on a body it changes the momentum of the body. The force is taken to be equal to the rate of change of the momentum. Thus if a force F acts on a body of weight m and changes its velocity from Vi to v2 in a time t, we have F = (mv2 — mv /t. The work done by the force is taken to be equal to the force multiplied by the distance through which it acts. 

The momentum of an object is its mass multiplied by its velocity. An object s momentum is directly related to the amount of energy it has. 

For a gas mixture at rest, the velocity distribution function is given by the Maxwell-Boltzmann distribution function obtained from an equilibrium statistical mechanism. For nonequilibrium systems in the vicinity of equilibrium, the Maxwell-Boltzmann distribution function is multiplied by a correction factor, and the transport equations are represented as a linear function of forces, such as the concentration, velocity, and temperature gradients. Transport equations yield the flows representing the molecular transport of momentum, energy, and mass with the transport coefficients of the kinematic viscosity, v, the thermal diffirsivity, a, and Fick s diffusivity, Dip respectively. 

The net momentum transport per molecule is thus the difference between the two velocities times m, or a/(2 a) Vin. By multiplying this by Zw) we find the total viscous force per unit area ... 

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. 

The current has the dimensions of a velocity and thus, when multiplied by the mass, it equals a momentum, Similarly, the time derivative of a velocity is an acceleration and hence one should obtain... 

Ehrenfest s first relation, the definition of the velocity and its associated momentum, is obtained using the commutator of H and the position vector r. This commutator, using the above Hamiltonian and multiplied by m, does indeed yield k as the momentum of an electron in an electromagnetic field... 

The structure of the expression for totai is that of a bilinear form it consists of a sum of products of two factors. One of these factors in each term is a flow quantity (heat flux q, mass diffusion flux jc, momentum flux expressed by the viscous stress tensor o, and chemical reaction rate rr)- The other factor in each term is related to a gradient of an intensive state variable (gradients of temperature, chemical potential and velocity) and may contain the external force gc or a difference of thermodynamic state variables, viz. the chemical affinity A. These quantities which multiply the fluxes in the expression for the entropy production are called thermodynamic forces or affinities. 

The first term on the LHS denotes the rate of accumulation of the velocity variance v - within the control volume. The second term on the LHS denotes the advection of the velocity variance by the mean velocity. The first term on the RHS denotes the production of velocity variance by the mean velocity shears. The momentum flux v vl is usually negative, thus it results in a positive contribution to variance when multiplied by a negative sign. The second term on the RHS denotes a turbulent transport term. It describes how variance is moved around by the turbulent eddies v. The third term on the RHS describes how variance is redistributed by pressure perturbations. This term is often associated with oscillations in the fluid (e.g., like buoyancy or gravity waves.) The fourth term on the RHS is called the pressure redistribution term. The factor in square brackets consists of the sum of three terms (i.e.,... 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are... 

First law The rate of change of linear momentum (mass of the body multiplied by the velocity of center of mass) is equal to the net impressed force on the body. 

Multiplying the mass flow M = AM/dt with the velocity w yields the momentum flux M-w. The momentum flux density is the momentum flux based on the cross-sectional area / ... 

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