Answer
See answers.
Work Step by Step
a. Assume that down is the positive direction. We are told k = 305 N/m, A = 0.280m, and m = 0.235 kg.
We calculate $\omega = \sqrt{k/m}=36.026\;rad/s$.
The mass has zero displacement and a positive velocity at t = 0, so the equation is a sine function.
$$y(t) = (0.280m)sin(36.0\;rad/s)(t))$$
b. The period of oscillation is $T=2 \pi/\omega = 0.1744\;s$.
The spring will first reach its longest extension at T/4. It will return there 1 period later, 2 periods later, etc.
$$t_{longest}=\frac{T}{4}+nT$$
Where n = 0, 1, 2, 3, …
The spring will first reach its longest extension at 3T/4. It will return there 1 period later, 2 periods later, etc.
$$t_{shortest}=\frac{3T}{4}+nT$$
Where n = 0, 1, 2, 3, …