What is the derivative of acceleration? What are its uses?

The derivative of position is the rate of change of position, which is velocity. The derivative of velocity is the rate of change of velocity, which is acceleration. The derivative of acceleration is the rate of change of acceleration, which is jerk. Also, in response to number 2, I don't think that acceleration has to be constant in mechanics. In fact, in real-life situations, I'd imagine that the many factors we'd have to consider would lead us to conclude that acceleration really ISN'T constant. You would just have to change your calculations a bit if it wasn't constant. Also, in regards to circular motion, imagine drawing tangent lines to a circle. The slope of the tangent line changes ---> Vector of acceleration changes. Acceleration at any one point in circular motion will change. Hope this helps :)

the time derivative of acceleration is called jerk. It has great importance. Just imagine a man sitting in a sports car. He pushes the accelerator at full and suddenly lifts his leg from the clutch. If the car does not jump, the man feels jerk. i.e we know that his acceleration would not be constant in such a case. You have yourself mentioned an example of a car so I thought to give another one!! :) You are correct in the details you added.! :)

it is called 'jerk' and i think it's important in designing rollercoasters, where a lot makes it more fun, and presumably other modes of transportation for a comfortable ride.

rate of acceleration

In mechanics, we study uniform acceleration (that is acceleration doesn't change). when you take the second derivative of the equation for trajectory, you are supposed to get a constant value, which is the acceleration. So taking its third derivative, which is the rate of change of acceleration, we will get zero. Because in physics, acceleration of system is always constant, and no change. Let say for sample the acceleration due to gravity d^2y/dt^2=-9.8 d^3y/dt^3=0 Edit: yeah I agree with you answerer 3. In real life acceleration is not constant. I'm pointing in mechanics only. we doesn't consider varying acceleration. In mechanics, we only assume that acceleration is constant for easy calculation. In reality its not. I already finished physics mechanics and engineering mechanics but doesn't encounter a problem with varying acceleration. If our second derivative would be an expression in terms of time, let say d^2y/dt^2=-9.8t (it means that acceleration varies with time t) if we take its third derivative it would be d^3y/dt^3=-9.8, it may indicate the rate of change of the acceleration with respect to time.. (probably)