How would you solve this: In the search for the Higgs Boson, it was posited that the potnetial energy stored between two higgs particles is given by a 1 dimensional function of the distance between the two particles (x) as U(x) = c + ax2 − bx4 where a, b, c are constants. a) Find a function for the force as a function of position. b) what value of position (x) is the equilibrium position in terms of a and b.

Use the Levi-Civita tensor to prove the following vector identities: [(A x B) x (C x D)] = [(A x B) . D] C - [(A x B) . C] D A x (B x C) = (A . C) B - (A . B) C

In certain physical models, the nonhomogeneous term, or forcing term, g(t) in the equation ay" + by' + cy = g(t) may not be continuous, but have a jump discontinuity. If this occurs, a reasonable solution can still be obtained using the following procedure. Consider the following initial value problem. 240 if 0sts7T/6 y" + 4y' + 40y = g(t); y(0) = 0, y'(0) = 0, where g(t) = . if t> 7x/6 Complete parts (a) through (c) below. (a) Find a solution to the initial value problem for 0 sts 71/6. The solution for 0sts 7n/6 is y(t) = - 6 e -21 *cos 6t - 2 e - 2t sin 6t + 6 (Type an equation.) (b) Find a general solution for t> 71/6. The general solution for t> 7x/6 is y(t) = C, e-2 cos 6t + C, e -2t sin 6t (Type an equation. Do not use d. D. e, E, i, or I as arbitrary constants since these letters already have defined meanings.) (c) Now choose the constants in the general solution from part (b) so that the solution from part (a) and the solution from part (b) agree, together with their first…

A particle has γ=18,399. a)Calculate c-v in m/s. (I would have asked for 1 - v/c, making the answer dimensionless, but the system doesn't seem to take numbers that small. Gamma is chosen to make the particle extremely close to the speed of light.) If your calculator gives problems, you might want to solve the appropriate equation for c-v or c(1 - v/c) and use an approximation. b) In the previous problem, in a race to the moon, by 3/4ths the distance, light is one or ten meters ahead of the particle. We routinely approximate mass as zero, gamma as infinite, and speed as the speed of light. ("Massless particles" -- gamma and m have to be eliminated from the expressions. Light is a true massless particle.) If a massless particle has momentum 1,739 MeV/c, calculate its energy in MeV.

An experimentalist in a laboratory finds that a particle has a helical path. The position of this particle in the laboratory frme is given by r(t)= R cos(wt)i + R sin(wt)j + vztk R,vz, and w are constants. A moving frame has velocity (Vm)L= vzk relative to the laboratory frame. In vector form: A)What is the path of the partical in the moving frame? B)what is the velocity of the particle as a function of time relative to the moving frame? C)What is the acceleration of the particle in each frame? D)How should the accelerartion in each frame be realted?Does your answer to part c make sense?

(b) Give the physical significance of the divergence of a vector point function.

Calculate the divergences of cach of the following vectors: (a) v = 3k (b) v = r (c) = (4xz + y?) î + (12a² – 2²) ĵ + (xy – yz) k (d) ở = -yî+ x} (c) v = . -

Q-1CC)IFE igen value of matrix is d, the corresponding eigen vector is 1 for matrin F882 32 then a is,

(a) Check all of the following that are correct statements, where E stands for ymc². Read each statement very carefully to make sure that it is exactly correct. O At speeds close to the speed of light, kinetic energy is approximately equal to (1/2)mv². O The energy principle can be written AEsys Wsurr. %3D O The definition of work is W = |I. %3D O The definition K = E - mc² is valid even for speeds near the speed of light. O The definition of work is W = . O The definition of work is W = FxAx + F,AY + F;Az. %3D O The energy principle can be written Esys,f = Esys,i + Wsurr (b) An object with mass 120 kg moved in outer space. When it was at location its speed was 18 m/s. A single constant force N acted on the object while the object moved to location m. What is the speed of the object at this final location? final speed m/s %3D

Consider following particles for particle swarm optimization to solve optimization problem where optimal solution is on zero. Initialize velocities values as per your choice within the range of 0.1 to 0.9. the initial values of particles  is  12 before starting the PSO algorithm. P1=0.2                           P2=0.5               P3=.34                        P4=.45     P5=0.48                a)Compute the fitness of above particles using X2+X +1 functions, sort out the particle based on fitness, most optimal particle should be on top of the list.  b)Update velocity and position of each particle where w=0.7, C1=C2=1.4, rand1=1, rand2=1