# Given an acceleration function and velocity function, how do I determine when something is decelerating or accelerating?

Given an acceleration function and velocity function, how do I determine when something is decelerating or accelerating?

I understand that if velocity x acceleration = (+) then it is accelerating and (-) if it is decelerating, but must I only determine this with a graph or interval chart?

Does a positive acceleration mean speeding up?

Also, when calculation at what time a function is changing direction, would you find the zeroes of a position time graph?

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Because the word “accelerate” has a technical meaning in mathematics, and the acceleration of a thrown object is constant in that technical sense, I for one would be much happier if you said “slowing down” instead of “decelerating”, and “speeding up” instead of “accelerating”. You’re really talking about the derivative of the speed function, $s'(t)$, where $s(t)=|v(t)|$, $v$ being the velocity (first derivative of position). But this $s'$ has no good physical use, since it doesn’t fit into Newton’s Law $F=MA$. – Lubin Oct 7 '13 at 3:19

Something is accelerating when the acceleration function, $a(t)$, is positive. Something is decelerating when the acceleration function is negative.

Note, for position function $x(t)$ and velocity function $v(t)$:

$x'(t) = v(t)$ and $x''(t) = v'(t) = a(t)$.

Something changes direction when the slope of $x(t)$ changes signs. By this, the slope either goes from positive to negative, or negative to positive. You could also view this as $v(t)$, which is the slope of $x(t)$ crossing the horizontal axis.

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Ah, yes, my mistake. I corrected it now. I believe it should be correct. – David Oct 7 '13 at 3:10
Yes, I like it in the new form, so I deleted my critical comment. – Lubin Oct 7 '13 at 3:39

Let $f(x)$ be an acceleration function. Now, let $f'(x)$ be the derivative of this function. All points where $f'(x)$ is negative will be where there is deceleration, and positive otherwise.

To find when a function is changing, let $f'(x)$ be the derivative of our function. A critical point is when the derivative is equal to 0 or is undefined. Find such points by looking for wherein there are divergences or infinite/undetermined expressions, or setting the derivative equal to 0.

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