3.1 · Motion, units, and related rates

Velocity, speed, and acceleration

Position $\displaystyle s(t)$ differentiates to velocity $\displaystyle v(t)$, and velocity differentiates to acceleration $\displaystyle a(t)$. Speed is $\displaystyle |v(t)|$, so negative velocity describes direction rather than negative speed.

When $\displaystyle v$ and $\displaystyle a$ have the same sign, speed increases. When their signs differ, speed decreases. At an instant with either value zero, inspect nearby signs before describing an interval of motion.

Differentiate before substituting the snapshot

For a changing circle, $\displaystyle A=\pi r^2$ relates area and radius at every time. Differentiate this relationship with respect to time to obtain $\displaystyle \frac{dA}{dt}=2\pi r\,\frac{dr}{dt}$. Substitute the current radius only after differentiating.

Draw and label the geometry first. A fixed length differentiates to zero; a changing length needs a time derivative. Convert units before combining quantities, and state what sign your final rate represents.

$$\frac{dA}{dt}=2\pi r\frac{dr}{dt}$$

Choose the relationship that links the unknowns

For a right triangle with changing legs, begin with $\displaystyle x^2+y^2=L^2$. If $\displaystyle L$ is fixed, differentiation gives $\displaystyle 2x\,\frac{dx}{dt}+2y\,\frac{dy}{dt}=0$. Use the geometry to find any missing length at the requested instant.

A rate equation is local: it describes the specific snapshot after all variables are evaluated. A negative height rate means the height is decreasing, not that the height itself is negative.