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# Interpreting direction of motion from position-time graph

Given the position-time graph of a linear motion, we can interpret whether the moving object moves forward or backward (or right or left, depending on how we define the motion!).

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• So.. Why is it, that at the peaks and valley's of a 'smooth' wave like this, is the slope zero , but at a sharp bend, the slope literally doesn't exist..? What's the difference?
• It simply doesn't make sense that the slope exists at a cusp. Take |x| for example; would it be -1, or 1, or 0? Why would one of these slopes take precedence over the others? Whereas with a smooth curve, the limit of the derivative at that point actually approaches a single, reasonable value.
• how to write equation for acceleration vs constant speed
• @ it is shown that the object is moving backward. However isn't the object at rest at that point? I am a little lost
(1 vote)
• At that point we have reached our starting point again, but, as can be seen with the slope of the line at that point, we are still walking backwards. This can also be seen by the fact that 1 second after this point, we are behind the starting point.