The velocity v of a free falling object of mass m is determined by the differential equation mv′ = mg − kv2, where g is the gravitational constant, and k is a constant depending on the object geometry. Note that m > 0, g > 0, and k > 0. a. Draw the direction field for the given differential equation for nonnegative velocities. Based on the direction field, determine the behavior of v as t → [infinity] assuming that v(0) = v0 with v0 ≥ 0. Does the limit limt→[infinity] v(t) depend on v0? b. Sketch the solution for several different initial conditions, and describe changes to limt→[infinity] v(t) when i. m increases and ii. k increases.

come closer and become far apart and become steam/gas

speed of light in vacuum is 300,000 km/s

speed of light in other mediums is given by the expression v = c/n, where c is the speed of light in vacuum and n is the refractive index of medium.

for the first medium v = 200,000 km/s

  so, refractive index of medium = 300000/200000 = 1.5

    1.5 refractive index is usually for glass. so first medium is glass.

for the second medium v = 250,00 km/s

so, refractive index of medium = 300000/250000 = 1.2

    1.2 refractive index is usually for water. so second medium is water.

for the third medium v = 0 km/s

so, refractive index of medium = 300000/0 = ∞

    ∞ refractive index is usually for opaque mediums. so third medium is opaque medium.