Consider the initial value problem mx" + cx' + kx = F(t), x(0) = 0, x'(0) = 0 80 modeling the motion of a damped mass-spring system initially at rest and subjected to an applied force F(t) where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k Newtons per meter, and F(t) = 80 cos(8t) Newtons. Solve the initial value problem. -4t 3 x(t) = e 12 cos (81) - ½ cos(81)) - cos (8t) + sin (81) help (formulas) Determine the long-term behavior of the system (steady periodic solution). Is lim x(t) = 0? If it is, enter zero 0047 If not, enter a function that approximates x(t) for very large positive values of t. For very large positive values of t, x(t) sp(t)=0 help (formulas) ≈ x

show all the steps and make sure answer is correct

3. A projectile is fired with an initial velocity of 128 ft/s from a cannon at time t = 0 from a height of 32 ft above the ground and at an angle of 30° with the horizontal. (a) Determine: x '(0) and y'(0). (These values represent the horizontal and vertical components of the initial velocity vector.) (b) Use integration to determine the position equations: x(t) and y(t) for the projectile t seconds after being fired by assuming that the projectile is in freefall. Thus, x"(t)=0 ft/s² and y"(t)=-32 ft/s² for all t≥ 0. (Note: x"(t) and y"(t) are the horizontal & vertical components of the projectile's acceleration vector at time t.) (c) Determine the maximum height, H, of the projectile and its horizontal distance, D, from the launch site at the time that it achieves this height. (Hint: The maximal height is attained at the time, t*, at which y'(t*) = 0.) (d) Determine the range of the projectile, R. YA (0,32) - P. 32' 128 0=30° x'(0) y'(0) H R Pr=(x(t), y(t)) is the position of the…

(1 point) Consider the initial value problem my" + cy' + ky = F(t), y(0) = 0, y'(0) = 0 modeling the motion of a spring-mass-dashpot system initially at rest and subjected to an applied force F(t), where the unit of force is the Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is 30 if 0 ≤t≤/2, F(t) 0 ift > π/2. a. Solve the initial value problem, using that the displacement y(t) and velocity y' (t) remain continuous when the applied force is discontinuous. For 0 ≤ts/2, y(t) Fortπ/2, y(t) = 0 (-3/8)(e^(-21))cos(6t)-(1/8)(e^(-2t))sin(6t)+3/8 help (formulas) help (formulas) b. Determine the long-term behavior of the system. Is lim y(t) = 0? If it is, enter zero. If not, enter a function that approximates y(t) for very large positive values of t. t-∞ For very large positive values of t, y(t) ≈0 help (formulas)