For an object in damped harmonic motion with initial amplitude a, period 2π/ω, and damping constant c, find an equation that models the displacement y at time t for the following.
(a)
y = 0 at time t = 0
y= ?
(b)
y = a at time t = 0
y= ?

  see below

Step-by-step explanation:

We presume the damping constant is the opposite of the multiplier of time in the exponential term. Then the equations are ...

(a)  y = a·e^(-ct)·sin(ωt)

(b)  y = a·e^(-ct)·cos(ωt)

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These are the standard equations for simple harmonic motion assuming there is no driving function.

  a = initial amplitude*

  c = damping constant**

  ω = frequency of oscillation in radians per second

  t = time in seconds

* Of course, when y(0) = 0, the motion never actually reaches this amplitude because it is subject to decay before it can.

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** In electrical engineering, damping is often specified in terms of a time constant, the time it takes for amplitude to decay to 1/e (≈36.8%) of the original amplitude. If that time is represented by τ, then the exponential factor is e^(-t/τ).

its 69.420

step-by-step explanation:

7. d. 64

8. i. 32

9. c. 125

10. j. none of the above

11. d. 81

12. i. 216

13. e. none of the above

14. g. 240

15. e. none of the above

16. g. 32

17.   e. none of the above