Two carts, joined together with a spring-loaded mechanism between them, move along on an air track at constant speed. The carts have different inertia, and the heavier cart is in front of the lighter cart. Then the mechanism is released, after which the speeds of the carts relative to each other is four times their original speed relative to the ground. In what direction is the lighter cart moving after the mechanism is released?

Let v1, v2, w be three linearly independent vectors in r 3 . that is, they do not all lie on the same plane. for each of the following (infinite) set of vectors, carefully sketch it in r 3 , and determine whether or not it is a vector space (i. e., a subspace of r 3 ). explain your reasoning

The key to stability is feedback between the reservoir and the fluxes into and/or out of the reservoir. assume that the rate of outflow from a reservoir depends on the size of the reservoir according to the following relationship: outflow rate= k x (size of reservoir), where k is a constant. a reservoir of water has a volume of 5000 liters, and the rate of outflow at steady state is 25 liters per minute. what is k? (give both the numerical value and its units.) what is the residence time? what is the relationship between k and the residence time?

Glass is transparent to visibile light under normal conditions; however, at extremely high intensities, glass will absorb most of the light incident upon it. this works through a process known as multiphoton absorption. in this process, several photons are absorbed at the same time. if very intense light whose photons carry 2ev of energy is shined onto a material with a band gap of 4ev, that light can be absorbed through two-photon absorption, because two photons have the right amount of energy to bridge the band gap. what is the minimum number of photons of 800-nm light that are needed to equal or exceed the band gap of fused silica glass