The third rotational kinematic equation relates angular velocity, angular acceleration, and angular displacement without requiring time:

Here $\omega_i$ is the initial angular velocity, $\omega_f$ is the final angular velocity, $\alpha$ is the constant angular acceleration, and $\Delta\theta$ is the angular displacement.

Derivation

We derive this equation by eliminating $t$ from the first two rotational kinematic equations. From the first equation:

Solving for time:

Substitute this expression into the second rotational kinematic equation: