For motion with constant angular acceleration, the instantaneous acceleration equals the average acceleration over any time interval. This is defined as the ratio of the change in angular velocity to the elapsed time.

Consider an object rotating with an initial angular velocity $\omega_i$ at the start time $t_0 = 0$, that accelerates to $\omega_f$ after a time duration $t$,

$$\alpha = \frac{\omega_f - \omega_i}{t - 0}$$

To derive the equation for the final angular velocity, we rearrange this algebraic relationship. First, multiply both sides by $t$ to clear the denominator:

$$\alpha t = \omega_f - \omega_i$$

Next, add the initial angular velocity $\omega_i$ to both sides to isolate $\omega_f$:

This result is the first rotational kinematic equation.

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