Help explain how direction change relates to acceleration I was doing some simple harmonic motion problems and I came across this picture describing the position, velocity and acceleration of a linear oscillator.   At the moment in time when v is 0 the linear oscillator should not be moving, only changing directions.  I'm having a hard time understanding why the acceleration is the greatest at that time (according to these graphs), since there is no velocity change.  Is it because acceleration is only the difference in velocity at two different points in time and not one?  How exactly does the change in direction affect acceleration?
edit: I found another question that answered my question. haha.
 A: 
Is it because acceleration is only the difference in velocity at two different points in time and not one? 

I think you've basically hit on the answer to your question here.  Acceleration is the derivative of velocity with respect to time, which means it is the instantaneous rate that the velocity is changing with time.  Acceleration is a measure of how fast velocity is changing; it does not depend on the particular velocity at any one time.  So even though the linear oscillator may not be moving at a particular time, it is undergoing a high acceleration as it switches directions.
Perhaps another easy way to recognize this in this specific case is by recalling Newton's second law: $\textbf{F}=ma$, where $\textbf{F}$ is the force applied to an object, $m$ is its mass, and $\textbf{a}$ is its acceleration.  In the case of a block on a spring (a certain kind of linear oscillator), the spring will exert the most force on the block when the block is furthest away from equilibrium.  This is also the point at which the block is motionless (i.e. its velocity is zero).  Thus the highest acceleration will occur at zero velocity in the case of a linear oscillator.
Hope that cleared up your confusion!
A: The expression "change in direction" implies some sort of discontinuity in motion, where in the referenced graphs, there is none. One could easily choose a different frame of reference such that the oscillating object appears never to change direction, only periodically speed up and slow down. The fact that the velocity value at the point in question happens to undergo a sign change is nothing more than an artifact of perspective. When you look at the graph, you see that the point of maximum acceleration occurs at the point where the slope of the velocity curve is steepest. It's not a coincidence... acceleration is by definition, rate of change of velocity, which is expressed in graphical form by slope.
