1. ## Kinematics (Complex)

Question:
A particle moves on the positive x-axis. The particle is moving towards the origin $\displaystyle O$ when it passes through the point $\displaystyle A$, where $\displaystyle x=2a$, with speed $\displaystyle \sqrt{\left(\frac{k}{a}\right)}$, where $\displaystyle k$ is constant. Given that the particle experiences an acceleration $\displaystyle \frac{k}{2x^2} + \frac{k}{4a^2}$ in a direction away from $\displaystyle O$, show that it come instantaneously to rest at a point $\displaystyle B$, where $\displaystyle x=a$. Immediately the particle reaches B the acceleration changes to $\displaystyle \frac{k}{2x^2} - \frac{k}{4a^2}$ in a direction away from $\displaystyle O$. Show that the particle next comes instantaneously to rest at $\displaystyle A$.

My Problem with this question:
A particle moves on the positive x-axis. The particle is moving towards the origin $\displaystyle O$ when it passes through the point $\displaystyle A$, where $\displaystyle x=2a$, with speed $\displaystyle \sqrt{\left(\frac{k}{a}\right)}$, where $\displaystyle k$ is constant. Given that the particle experiences an acceleration $\displaystyle \frac{k}{2x^2} + \frac{k}{4a^2}$ in a direction away from $\displaystyle O$, show that it come instantaneously to rest at a point $\displaystyle B$, where $\displaystyle x=a$. Immediately the particle reaches B the acceleration changes to $\displaystyle \frac{k}{2x^2} - \frac{k}{4a^2}$ in a direction away from $\displaystyle O$. Show that the particle next comes instantaneously to rest at $\displaystyle A$.