AHL 5.13 (HL)—Kinematics

For position s(t), velocity v(t)=ds/dt and acceleration a(t)=dv/dt=d²s/dt². Integrating velocity or acceleration needs initial conditions to recover the earlier quantity.

The sign depends on the chosen positive direction. A negative velocity means motion opposite that direction; a negative acceleration does not necessarily mean the object is slowing if velocity is also negative. Keep the sign convention fixed throughout the interval.

If s(t)=t³−6t²+9t, then v=3t²−12t+9 and a=6t−12. At t=1, v=0 but a=−6, so the object is instantaneously stationary while its velocity is changing.

Distance is not signed displacement. Use |v| or integrate speed for total distance, and use the stated time interval when locating turning points.

Also use $a=v\,dv/ds$ when velocity is given as a function of displacement. Over $[t_1,t_2]$, displacement is $\int_{t_1}^{t_2}v(t)\,dt$ while total distance is $\int_{t_1}^{t_2}|v(t)|\,dt$; split at zeros of $v$. Dot notation $\dot x$ and $\ddot x$ denotes first and second time derivatives.