Consider the initial value problemmy" + cy' + ky = F(t), y(0) = 0, y'(0) = 0modeling 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 theNewton (N). Assume that m2 kilograms, c= 8 kilograms per second, k = 80 Newtons per meter, and the applied force in Newtons is30if 0 T/2.a. Solve the initial value problem, using that the displacement y(t) and velocity y'(t) remain continuous when the applied force isdiscontinuous.For 0 T/2, y(t) = e^(-2t)((-1/3-1/3e^pi)cos(6t)+(-2/15-(2/15)e^pi)sin(6t))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) forvery large positive values of t.help (formulas)For very large positive values of t, y(t)

Answered: Consider the initial value problem my"…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Block 1 of mass m₁ 0.5 kg slides on frictionless surface with velocity 8 kg. Block 1 bounces back with V₁ 7 m/s and hits block 2 of mass m2 = velocity of = = 3 m/s. Block 2 is fixed to a spring and begins at equilibrium. The spring has spring constant k 20 N/m. If the collision occurs instantaneously, how far does block 2 move from its initial position? = m1 1 = V m1 1 m2 2 lo w 7₁ m2 H 2 WWW
2. A ball is dropped from a 20-foot building with a constant acceleration due to gravity of 9.8m/s?. Show the particular solution of s(t). From the initial point(t=0): S=20 ft V=0
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y" + 12y' + 36y= 0
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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)

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