**Time Rates**

If a quantity *x* is a function of time *t*, the time rate of change of *x* is given by *dx*/*dt*.

When two or more quantities, all functions of *t*, are related by an equation, the relation between their rates of change may be obtained by differentiating both sides of the equation with respect to *t*.

**Basic Time Rates**

- Velocity, $v = \dfrac{ds}{dt}$, where $s$ is the distance.

- Acceleration, $a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2}$, where $v$ is velocity and $s$ is the distance.

- Discharge, $Q = \dfrac{dV}{dt}$, where $V$ is the volume at any time.

- Angular Speed, $\omega = \dfrac{d\theta}{dt}$, where $\theta$ is the angle at any time.

- Angular Acceleration, $\alpha = \dfrac{d\omega}{dt} = \dfrac{d^2\theta}{dt^2}$, where $\omega$ is the angular velocity and $\theta$ is the angular displacement.

**Steps in Solving Time Rates Problem**

- Identify what are changing and what are fixed.
- Assign variables to those that are changing and appropriate value (constant) to those that are fixed.
- Create an equation relating all the variables and constants in Step 2.
- Differentiate the equation with respect to time.