Topic: Oscillatory Motion
Concepts: Mass-spring system, Vectors

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Solution:

Since $\vec{r}=x\vec{i}+y\vec{j}$, we have $\vec{F}=-k\vec{r}$ so we can identify the spring constant as $k=8\,\mathrm{N/m}$. The period of a mass-spring system is given by

The particle first returns to the origin in half the period,

so the answer is C.

Topic: Energy
Concepts: Conservation of energy, Spring potential energy

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Solution:

From the previous problem, we have the spring constant $k=8\,\mathrm{N/m}$. The particle is launched with speed $v=\sqrt{(3\,\mathrm{m/s})^2+(4\,\mathrm{m/s})^2}=5\,\mathrm{m/s}$. By conservation of energy,

so the answer is B.