Differentiating to get v and a
Rates of change link the three quantities. The velocity is the rate of change of displacement, and the acceleration is the rate of change of velocity. So we differentiate once to go from s to v, and again to go from v to a.
Key formula /
v = ds/dt and a = dv/dt = d²s/dt². Maximum or minimum displacement occurs when v = 0; maximum or minimum velocity occurs when a = 0.
Turning points of motion
Since v = ds/dt, the displacement is greatest or least where v = 0 (the particle is momentarily at rest). Likewise the velocity is greatest or least where a = 0. This is exactly the idea of a turning point applied to motion.
The second derivative d²s/dt² measures how quickly the velocity itself is changing. A positive acceleration speeds a particle up when it is already moving in the positive direction, while a negative acceleration slows it down or drives it backwards. Checking the sign of a at a time when v = 0 tells you whether that instant gives a maximum or a minimum displacement, just as the second-derivative test does for an ordinary curve.
Worked example
Given s = t³ − 6t² + 9t, differentiate: v = 3t² − 12t + 9 and a = 6t − 12. The particle is at rest when v = 0: 3(t² − 4t + 3) = 0, so t = 1 or t = 3 s. At t = 1, a = 6 − 12 = −6 m s⁻². At t = 2, v = 12 − 24 + 9 = −3 m s⁻¹.
Remember
- Differentiate s → v → a.
- Max/min displacement: v = 0.
- Max/min velocity: a = 0.