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I know this may seem a weird question, but it always bothers me. My physics book (Resnick,Halliday,Walker), and also various sites never say anything beyond acceleration.

But when a moving body is being acted by a variable force , its acceleration will definitely change: it will either increase or decrease. Then there will be rate of change of acceleration with respect to time. So, why don't books mention this? What is the cause for not measuring $\frac{d\vec{a}}{dt}$ ? If it exists, what is the use of it?

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Your question is not weird; it is legitimate. It is possible, it exists, can be of use and it is called jerk, jolt, surge or lurch, and is defined by any of the following equivalent expressions: $$\vec j(t)=\frac {d\vec a(t)} {dt}=\dot {\vec a}(t)=\frac {{d}^2 \vec v(t)} {dt^2}=\ddot{\vec v}(t)=\frac {{d}^3 \vec r(t)} {dt^3}=\overset{...}{\vec r}(t)$$

It is useful in the Dirac-Lorenz equation (as Emilio linked).

In case you are asking yourself, a fourth derivative (rate of jerk) is also defined, and it is called jounce

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    $\begingroup$ I've heard it referred to as "impetus". Having ridden on vehicles for which I've implemented motor-control code, it can be surprisingly perceptible. If a vehicle's velocity is linearly ramped to zero and then stays there, the transition from decelerating to stopped will be very noticeable almost no matter how gradual the deceleration was. A more rapid deceleration can feel smooth if one limits the rate of change of acceleration. $\endgroup$
    – supercat
    Commented Sep 23, 2014 at 20:26
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    $\begingroup$ @supercat: I've only ever heard of impetus as another term for momentum. $\endgroup$
    – Kyle Kanos
    Commented Oct 3, 2014 at 13:59
  • $\begingroup$ @KyleKanos: That's possible, and would make sense. The fact that I've simply heard the term used is why I wrote a comment instead of an answer. I wasn't sure the usage was legitimate, and if it's isn't that's good to know. I wonder if the upvote on my comment was because of my mention of riding vehicles with programmed acceleration characteristics. I would that that would be the sort of thing from which a college physics program could benefit even if the "vehicle" was little more than wheels and a seat and couldn't go more than a couple meters per second. $\endgroup$
    – supercat
    Commented Oct 3, 2014 at 16:57
  • $\begingroup$ I heard from an engineer working in drone-control programming that they have to consider the 4th, 5th and even 6th time-derivatives of displacement. They refer to them as snap, crackle and pop. $\endgroup$
    – Oscar Bravo
    Commented Feb 15, 2019 at 7:40
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This does exist, and it is called Jerk, see the Wikipedia page on jerk.

It is used quite frequently in physics concerning humans, as we are able to sense this, and there are limits to how much jerk a human can endure.

It is, however, quite abstract and therefore more difficult to comprehend, which might be the reason that lower level textbooks do not mention it.

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There are many reasons why acceleration would not be constant. Books often don't mention it because they are getting mathematicians used to the concept of first derivatives before moving onto second, third etc...

Consider the following example which ends up leading to Tsiolkocsky's Rocket Equation: http://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation

A rocket has mass, and propellant. To accelerate the rocket expends propellant over time at a fixed rate, giving a constant force. This means the Mass changes over time at a fixed rate, and if you look at f=ma, you quickly realize that acceleration is not constant, as acceleration becomes a function of a variable mass.

In this case total mass is a second derivative of delta-v (total change in velocity).

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  • $\begingroup$ This is a good common-place example of jerk (increasing acceleration). You can see and hear it when you launch a firework rocket at Guy Fawkes Night (or 4th of July, or quatorze juillet, or Chinese New Year or whenever). The rocket scoots off and the whizz increases in frequency as it ascends. $\endgroup$
    – Oscar Bravo
    Commented Feb 15, 2019 at 7:37