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 isthe Newton (N). Assume that m = 2 kilograms, c = 8 kilograms per second, k = 80 Newtons per meter, and F(t) = 30e * Newtons.a. Solve the initial value problem.y(t) =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 approximatesy(t) for very large positive values of t.For very large positive values of t, y(t) ~help (formulas)

The given initial value problem is:my''+cy'+ky=F(t); y(0)=0, y'(0)=0

Answered: Consider the initial value problem my"…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Let the position at time t seconds of a particle moving in a coordinate line be given by s(t) = cos(3t) centimeters.(a) Find its velocity v(t) = s0(t) at time t.(b) Show that 9s(t)2 +v(t)2 = 9.(c) Find its acceleration a(t) = v0(t) = s00(t) at time t.
= (4) Suppose the acceleration of an object is given by a(t) = t m/s². Its initial velocity is v(0) = 5 m/s. (a) Find a general antiderivative for a(t). (That is, your answer should contain +C.) (b) Use the information that v(0) = 5 to solve for C. (c) Use your answers to parts (a) and (b) to write a formula for v(t) that has no +C in it. (d) Use your formula from part (c) to compute v(3).
A mass weighing 16 pounds stretches a spring feet. The mass is initially released from rest from a point 4 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a 1 damping force that is numerically equal to - the instantaneous velocity. Find the equation of motion x(t) if the mass is driven by an external force equal to f(t) = 20 cos(3t). (Use g = 32 ft/s? for the acceleration due to gravity.) x(t) = ft
A particle moves on the 0x axis with acceleration proportional to velocity. Assuming that v(0) = 3, v(1) = 2 and x(0) = 0, determine the position of the particle at time t.
A mass weighing 16 pounds stretches a spring 8 feet. The mass is initially released from rest from a point 4 feet below the equilibrium position, and the subsequent motion takes place in a medium that offers a 1 damping force that is numerically equal to - the instantaneous velocity. Find the equation of motion x(t) if the mass is driven by an external force equal to f(t) = 20 cos(3t). (Use g = 32 ft/s2 for the acceleration 2 due to gravity.) x(t) = ft
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…

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