Initial velocities

If p is taken to be a constant during a skid, application of Amontons law leads to a very simple relationship between the initial velocity of the vehicle and the length of the skid mark. The initial kinetic energy is mv fl, and this is to be entirely dissipated by the braking action, which amounts to a force F applied over the skid distance d. By Amontons law. 

Thus if Amontons law is obeyed, the initial velocity is determined entirely by the coefficient of friction and the length of the skid marks. The mass of the vehicle is not involved, neither is the size or width of the tire treads, nor how hard the brakes were applied, so long as the application is sufficient to maintain skidding. 

Here, the symbol denotes an averaging over the flux-weighted distribution [7, 8] for positive or negative initial velocities of the reaction coordmate. In figure A3.8.2 is shown the correlation fimction K(t) for the particular case of the reaction of methyl vinyl ketone with cyclopentadiene in water. The leveling-off of this... 

Figure Bl.7.17. (a) Schematic diagram of a single acceleration zone time-of-flight mass spectrometer, (b) Schematic diagram showing the time focusing of ions with different initial velocities (and hence initial kinetic energies) onto the detector by the use of a reflecting ion mirror, (c) Wiley-McLaren type two stage acceleration zone time-of-flight mass spectrometer.

Ions generated in the ion source region of the instrument may have initial velocities isotropically distributed in tliree dimensions (for gaseous samples, this initial velocity is the predicted Maxwell-Boltzmaim distribution at the sample temperature). The time the ions spend in the source will now depend on the direction of their initial velocity. At one extreme, the ions may have a velocity Vq in the direction of the extraction grid. The time spent in the source will be... 

At the other extreme, ions with initial velocities in the direction opposite to the accelerating potential must first be turned around and brought back to their initial position. From this point their behaviour is the same as described above. The time taken to turn around in the ion source and return to the initial position is given by ... 

This reduces many of the complications arising from the need for spatial focusing. Also, the initial velocity of ions generated are invariably in the TOP direction. 

Molecular beam sample introduction (described in section (Bl.7.2)). followed by the orthogonal extraction of ions, results in improved resolution in TOP instruments over eflfrisive sources. The particles in the molecular beam typically have translational temperatures orthogonal to the beam path of only a few Kelvin. Thus, there is less concern with both the initial velocity of the ions once they are generated and with where in the ion source they are fonned (since the particles are originally confined to the beam path). 

It is appropriate to consider first the question of what kind of accuracy is expected from a simulation. In molecular dynamics (MD) very small perturbations to initial conditions grow exponentially in time until they completely overwhelm the trajectory itself. Hence, it is inappropriate to expect that accurate trajectories be computed for more than a short time interval. Rather it is expected only that the trajectories have the correct statistical properties, which is sensible if, for example, the initial velocities are randomly generated from a Maxwell distribution. 

The algorithm was applied to the MD simulations of a box of water molecules. The three-center water model was used [23]. The initial positions were at the equilibrium therefore all displacements were zero. The initial velocities were... 

IlyperChem can either use initial velocilies gen eraled in a previous simulation or assign a Gaussian distribution of initial velocities derived from a random n iim her generator. Random numbers avoid introducing correlated motion at the beginn ing of a sim illation. ... 

The Restart check box can be used in ctiii junction with the explicit editing of a IIIX file to assign completely user-specified initial velocities. This may be useful in classical trajectory analysis of chemical reactions where the initial velocities and directions of the reactants are varied to statistically determine the probability of reaction occurring, or n ot, in the process of calculating a rate con -Stan t. 

Coordin ates of atom s can he set by n orm al translation orrotation of HyperCh cm molecules, fo set initial velocities, however, it is necessary to edit th e H l. file explicitly. The tin it o f velocity in the HIN file is. An gstrom s/picosecon d.. Areact.hin file and a script react.scr are in eluded with HyperChem to illustrate one simple reacting trajectory. In order to have these initial velocities used in a trajectory the Restart check box of the Molecular Dynamics Option s dialog box must he checked. If it is n ot, the in itial velocities in the HIN file will be ignored and a re-equilibration to the tern peratiire f of th e Molecular Dyn am ics Option s dialog box will occur. This destroys any imposed initial conditions on the molecular dynamics trajectory. 

The initial velocities may also be chosen from a uniform distribution or from a simp Gaussian distribution. In either case the Maxwell-Boltzmann distribution of velocities usually rapidly achieved. 

The initial velocities are often adjusted so that the total momentum of the system is zer Such a system then samples from the constant NVEP ensemble. To set the total line momentum of the system to zero, the sum of the components of the atomic momen along the x, y and z axes is calculated. This gives the total momentum of the system each direction, which, when divided by the total mass, is subtracted from the atom velocities to give an overall momentum of zero. 

Having set up the system and assigned the initial velocities, the simulation proper a commence. At each step the force on each atom must be calculated by differentiating ll potential function. The force on an atom may include contributions from the varioi... 

If the masses are displaced in an arbiPary way or arbiPary initial velocities are given to them, the motion is asynchronous, a complex mixture of synchronous and antisynchronous motion. But the point here is that even this complex motion can be broken down into two normal modes. In this example, the synchronous mode of motion has a lower frequency than the antisynchronous mode. This is generally Pue in systems with many modes of motion, the mode of motion with the highest symmePy has the lowest frequency. 

As described previously, the Leap-frog algorithm for molecular dynamics requires an initial configuration for the atoms and an initial set of velocity vectors. /2- These initial velocities can come... 

Statistically, in a high-pressure region, an ion will be struck by neutral molecules randomly from all angles. The ion receives as many collisions from behind as in front and as many collisions from one side as from the other. Therefore, it can be expected that the overall forward motion of the ion will be maintained but that the trajectory will be chaotic and similar to Brownian motion (Figure 49.4b). Overall, the ion trajectory can be expected to be approximately along the line of its initial velocity direction, since it is still influenced by the applied potential difference V. 

Therefore, in the RF mode, ions transmitted through the rod guide are subjected to (1) an oscillation in step with the variations of the RF field in the x,y-plane, (2) a drift or guided motion caused by the inhomogeneity of the RF field (x,y-plane), and (3) a forward motion (z-direction) due to any initial velocity of the ions on first entering the rod assembly. The separate motions... 

Several references quote a length of 100 nozzle diameters for the length of the estabhshed flow region. However, this length is dependent on initial velocity and Reynolds number. 

Example 10 Response to Instantaneous Valve Closing Compute the wave speed and maximum pressure rise for instantaneous valve closing, with an initial velocity of 2,0 m/s, in a 4-in Schedule 40 steel pipe with elastic modulus 207 X 10 Pa, Repeat for a plastic pipe of the same dimensions, with E = 1.4 X 10 Pa. The liquid is water with P = 2.2 X 10 Pa and p = 1,000 kg/m. For the steel pipe, D = 102,3 mm, b = 6,02 mm, and the wave speed is... 

Only a small frac tion of U is available to provide kinetic energy to the fragments. There are few data available, but in five incidents analyzed by High (unpublished data), no fraction was greater than 0.15. The fragment initial velocity can be assessed from 0.15 U = V hLVf. 

Newton s equation of motion is a second-order differential equation that requires two initial values for each degree of freedom in order to initiate the integration. These two initial values are typically a set of initial coordinates r(0) and a set of initial velocities v(0). ... 

Unlike the initial coordinates, which can be obtained experimentally, the only relevant information available about atomic velocities is the system s temperature T, which determines the velocity distribution. In the absence of a better guideline, initial velocities (v , Vy, Vy) are usually randomly assigned from the standard Maxwellian velocity distribution at a temperature T,... 

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