Consider an ideal gas of particles with mass m. the probability that a particle is moving at a speed v is proportional to exp(-mv2kt), where v is the magnitude of the three-dimensional velocity vector. (hint: you can use any coordinate system to do these calculations, but be careful to perform the change of variables properly, including the correct differential element if you use spherical or cylindrical polar coordinates.] (a) determine the normalized probability that a particle is moving with speed v, where v is the magnitude of the three-dimensional velocity vector. (b) determine the average speed of the particles, where the speed v is the magnitude of the three- dimensional velocity vector. (c) determine the average kinetic energy of the particles. (d) determine the average value of vxy, where vx represents the magnitude of the component of the velocity in the x-y plane.

so here is the information for current research:

-nrel's geothermal analysts, scientists, and engineers perform a full range of research from developing advanced methods of locating and accessing geothermal resources to exploring innovative technologies to extract and produce clean energy from the heat of the earth

and here is the information for the technology of geothermal energy:

-it's clean and sustainable

-resources of geothermal energy range from the shallow ground to hot water and hot rock found a few miles beneath the earth's surface, and down even deeper to the extremely high temperatures of molten rock called magma

-almost everywhere, the shallow ground or upper 10 feet of the earth's surface maintains a nearly constant temperature between 50° and 60°f (10° and 16°c)

-geothermal heat pumps can tap into this resource to heat and cool buildings

-a geothermal heat pump system consists of a heat pump, an air delivery system (ductwork), and a heat exchanger-a system of pipes buried in the shallow ground near the building

-in the winter, the heat pump removes heat from the heat exchanger and pumps it into the indoor air delivery system

-in the summer, the process is reversed, and the heat pump moves heat from the indoor air into the heat exchanger

-the heat removed from the indoor air during the summer can also be used to provide a free source of hot water

the answer is a. hope this .

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