Acceleration of acceleration?
Hi everyone.
I was wondering about this. If the position of an object changes in time, the object has velocity. If the velocity of an object changes in time, the object is accelerating (decelerating) If the acceleration of an object changes in time, could we hypothetically have acceleration of acceleration. I have this scenario in my head: An asteroid is at rest, far away from earth. The earth starts dragging the asteroid towards it. The asteroid will accelerate towards earth ever so slightly (since gravity depends on distance) The closer it gets to earth, not only will it go faster, but it will also accelerate faster. So we will have the change of acceleration in time: [itex]a\prime = \lim_{\Delta t\to\ 0}\frac{\Delta a}{\Delta t}[/itex] So acceleration of acceleration would be in [itex]\frac{m}{s^3}[/itex] or rather [itex]\frac{\frac{m}{s^2}}{s}[/itex] if it appeals more. Would this be useful? We could calculate the exact acceleration in any given moment as opposed to having the average acceleration. But that's beside the point since we could always calculate the exact acceleration if we know how far it is from a planet. But if we needed to know the exact speed of the asteroid after let's say 20 hours, we would get an incorrect answer if we treated the acceleration as if it were constant. So [itex]a_1 = a_0 \pm a\prime t[/itex] so if [itex]a_0 = 0[/itex] then [itex]a_1 = a\prime t[/itex] and if [itex]v_1 = v_0 \pm at[/itex] and if [itex]v_0 = 0[/itex] then [itex]v_1 = at[/itex] => [itex]v_1 = a\prime t^2[/itex] if the object is starting to move from rest Does any of this make sense? Waiting for someone to point out a flaw in this. 
Re: Acceleration of acceleration?
I didn't read through but a' is called the "jerk".

Re: Acceleration of acceleration?
After "jerk", the next three derivitives are affectionately known as "snap", "crackle" and "pop".
Ordinarily one does not attack these kinds of problems by looking at higher and higher order derivitives though. Instead one writes down a differential equation that relates, for instance, acceleration to velocity and position. One can use differential calculus to attempt to solve such an equation, reducing it to a form that expresses position as a function of time. Failing that, there are computational methods (such as Runge Kutta) that generate approximate solutions by running a kind of simulation and advancing stepwise. Such approaches often treat the second derivitive (acceleration) as a variable and work in part by estimating its average value over the duration of each small step. This sounds similar to what you are talking about. 
Re: Acceleration of acceleration?
You can do a simple experiment to 'feel' this when braking your automobile.
As you get it almost stopped, lighten pressure on the brake so you don't slam back into seat when it stops, instead making a gradual approach to rest. Acceleration with a negative sign is the force pushing you forward. Your passengers will appreciate your awareness of and attention to "jerk". 
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