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If $a(t)$ denotes the acceleration of an object, then $a^\prime(t)$ represents the jerk.

I'm looking for an intuitive explanation of this phenomena. I'm hoping the following anecdote provides the intuition of how one might experience / explain jerk.

Suppose you're driving your car, and you approach a stoplight. The car then has a non-constant change in acceleration, $a'(t).$ Now, the moment the car comes to a full stop, we experience the car rocking back and forth. This is because the acceleration $a(t)$ goes to zero, but the change in acceleration at that time $a'(t)$ is . . .

Now, I would attempt to finish the explanation, but I lack the requisite language. My intuition is that $a'(t)$ is large, infinite, or is best approximated by an impulse function as $a(t)$ gets close to $0$. I'm not assuming that $a'(t)$ is differentiable, nor am I assuming it is even continuous.

Is this the right intuition? Would one of you be willing to provide another anecdote that one might experience on a day to day basis?

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    $\begingroup$ An electric shuttle bus at an airport is a great demonstration of jerk and all bus drivers there are jerk experts. $\endgroup$ – safesphere Oct 10 '18 at 19:37
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Acceleration changes with force, so the derivative of acceleration changes with the derivative of force. In other words, if,

$$ m\ddot x = F, $$ then, $$ m\dddot x = \dot F. $$

So, jerk is the rate at which the force is changing, divided by the object's mass. If you "jerk" an object you are briefly changing the applied force from zero to some number, and the shorter the time period over which you bring it from no force to full force, the higher the "jerk" of the action.

For a real-world example, consider the velocity of your hand on a throttle. If your rocket engine produces a thrust proportional to the throttle's position, then the speed of your hand is proportional to the jerk your passengers will experience.

Another intuitive aid may be thinking about acceleration in terms of the gravity that is's equivalent to. If you are a passenger on a rocket or an elevator and you feel your weight suddenly double, that's high-jerk. If your weight is slowly increasing or decreasing, it is low-jerk. This is because the weight you feel goes up and down proportionally to your acceleration.

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