Velocity, Acceleration and Jerk

If $s=f(t)$ is the displacement of a particle moving along a straight line, then $f'(t)$ is the rate of change of displacement with respect to time — in other words, $f'(t)$ is the particle's velocity. The particle's speed is the absolute value of the velocity, $|f'(t)|$ (speed is never negative, even when the velocity is, which happens when the particle moves in the negative direction).

The rate of change of velocity with respect to time is called the acceleration, denoted $a(t)$. Since acceleration is the derivative of velocity, and velocity is itself the derivative of displacement, acceleration is the second derivative of the displacement function:

Going one step further, the third derivative of the displacement function, $s'''(t)=v''(t)=a'(t)$, is called the jerk, denoted $j$:

The jerk measures the rate of change of acceleration — it is named for the abrupt, jolting sensation caused by a sudden change in acceleration (for instance, in a vehicle that brakes or accelerates suddenly).

Worked Example 1 — Constant acceleration. A car's distance is $s=4t^2+3t$ (meters, seconds). Velocity: $v=\dfrac{ds}{dt}=8t+3$. Acceleration: $a=\dfrac{dv}{dt}=8$ (a constant — this is uniformly accelerated motion, independent of time). At $t=20$: velocity $=8(20)+3=163$ m/sec, and acceleration $=8$ m/sec² (unchanged, since it doesn't depend on $t$). …