Given an acceleration function and velocity function, how to determine whether the particle is decelerating or accelerating?

Question

Given an acceleration function and a velocity function, how do I determine whether the particle is decelerating or accelerating?

I understand that if velocity $\times$ acceleration is positive then it is accelerating, and if negative then 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?

Answer 1

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.

Answer 2

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.

Answer 3

I understand that if velocity $\times$ acceleration is positive then it is accelerating, and if negative then it is decelerating

Echoing Lubin's comment:

• if velocity and acceleration are in the same direction, please say speeding up instead of "accelerating";
• if velocity and acceleration are in opposite directions, please say slowing down (or retarding; but do note that this refers to decreasing speed, not decreasing velocity) instead of "decelerating" (this is a bad, ambiguous word; avoid it).

Because when the particle's velocity is negative:

• when the particle is speeding up, its acceleration is negative, so saying that it is "accelerating" is either not meaningful or falsely suggesting that its acceleration is positive;
• when the particle is slowing down, its acceleration is positive, so saying that it is "decelerating" may give the wrong impression that its acceleration is negative.

Does a positive acceleration mean speeding up?

In the previous bullet point whose particle has a positive acceleration, its acceleration and velocity having different signs means that the net force acting on the particle is opposite in direction to its motion; that is, the particle is slowing down.