SL 5.9—Kinematics

Kinematics links position, velocity and acceleration by differentiation.

If s(t) is position, v=s′ and a=v′=s″; signs depend on the chosen positive direction.

Example

For s=4t²−t, v=8t−1 and a=8, so velocity changes while acceleration stays constant.

State the time interval and direction convention before interpreting motion.

Negative velocity means motion opposite the chosen direction, not necessarily slowing down.

From $t_1$ to $t_2$, displacement is $\int_{t_1}^{t_2}v(t)\,dt$, while total distance is $\int_{t_1}^{t_2}|v(t)|\,dt$. Split the distance integral wherever $v(t)=0$. Speed is $|v|$: an object slows when velocity and acceleration have opposite signs and speeds up when their signs agree.