Find the solution to the Initial Value Problem describing the horizontal motion of a mass attached to a spring, mx = −k(x − 1), x(0) = 4, x(0) = 0, for m = 1, k = 9,1 = 4. Describe in words the motion of the mass.

The density of air changes with height. Under some conditions density p, depends on height z, and temperature T according to the following equation where Po and A are both constants. A meteorological balloon ascends (i.e., starts at z = 1 and gains height) over the course of several hours. Complete parts (a) and (b) below. Az P(z,T) = Po e ..... dz v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, (a) Assuming that the balloon ascends at a speed v (i.e., dt as measured by the weather balloon. Choose the correct answer below. dp Az dT O A. dt T2 dt dp %3D dt Az dT dp С. dt %3D + T dt dp Az) dT O D. dt T2) dt (b) Assume that v = 1, Po = 1, and A = 1 and that when t= 0, T= 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = 0, and solve this…

5. A ball is launched vertically into the air from a height of h meters and with an initial upward velocity of u meters/second. The ball's height above ground is given by the equation H(t) = -4.9t2 + vt + h, where H is in meters and t is in seconds. (This is the metric version of the gravity model.)

(1) A mass weighing 4 lb stretches a spring 2 in. Suppose that the mass us displaced an additional 6 in. and then released. The mass is in a medium that exerts a viscous resistance of 6 lb when the mass has a velocity of 3 ft/sec. Set up the initial value problem that describes the motion of the mass.