Roughly speaking, the second derivative measures how the rate of change of a quantity is itself changing; for example, the second derivative of the position of a vehicle with respect to time is the instantaneous acceleration of the vehicle, or the rate at which the velocity of the vehicle is changing with respect to time.
12.
This slope therefore defines the average acceleration over the interval, and reducing the interval infinitesimally gives \ begin { matrix } \ frac { dv } { dt } \ end { matrix }, the instantaneous acceleration at time t, or the derivative of the velocity with respect to time ( or the second derivative of the position with respect to time ).
13.
If we make two measurements in a small enough time interval then we can * approximate * the instantaneous velocity of the object, and if we make more measurements in a small enough time interval then we can * approximate * its instantaneous acceleration, jerk etc . But I don't know of any way to actually * measure * instantaneous velocity, acceleration, jerk etc in a general reference frame.
