Aparticle’s velocity (in m/s) in a gas is a random variable (let it be v ) with pdf fv (v) = av2 e −bv, v > 0, where b is a constant that depends on the temperature of the gas and the mass of the particle. answer the following questions. (a) what should a be for this to be a valid pdf as a function of b? (b) what is the probability the velocity of the particle is between 10 and 100 meters per second? (c) the kinetic energy of a particle is w = 1 2mv 2 , where v is the speed. what is the pdf of w? (d) what is the mean kinetic energy of a particle as a function of constant b?

Aparticle of mass m is projected with an initial velocity v0 in a direction making an angle α with the horizontal level ground as shown in the figure. the motion of the particle occurs under a uniform gravitational field g pointing downward. (a) write down the lagrangian of the system by using the cartesian coordinates (x, y). (b) is there any cyclic coordinate(s). if so, interpret it (them) physically. (c) find the euler-lagrange equations. find at least one constant of motion. (d) solve the differential equation in part (c) and obtain x and y coordinates of the projectile as a function of time. (e) construct the hamiltonian of the system, h, and write down the hamilton’s equations (canonical equations) of motion.

Suppose a wire with a current is pushed upward by a magnetic field. how would the wire move if the direction of the current reversed? explain your answer.

Use the illustration to describe how the electromagnetic spectrum changes as its frequency moves from radio waves to higher energy gamma waves in terms of wavelength and amplitude.