The formula ωf = ωi + αt can be derived from the basic definitions of angular displacement, angular velocity, and angular acceleration. Let's go through the derivation step by step.

1. Angular displacement (θ) is the angle through which an object rotates in a given time. It is the difference between the initial angle (θi) and the final angle (θf):

   `θ = θf - θi`

2. Angular velocity (ω) is the rate of change of angular displacement with respect to time:

   `ω = dθ/dt`

   We have both initial angular velocity (ωi) and final angular velocity (ωf) for the given time interval.

3. Angular acceleration (α) is the rate of change of angular velocity with respect to time:

   `α = dω/dt`

   Now, we need to find the relationship between ωf, ωi, α, and time (t).

4. From the definition of angular acceleration, we can write:

   `α = (ωf - ωi) / t`

5. Rearranging the equation to isolate ωf, we get:

   `ωf = ωi + αt`


This is the formula we were looking to derive. It shows the relationship between the initial and final angular velocities, angular acceleration, and time. This formula is analogous to the linear motion equation `vf = vi + at`, where `vf` is the final linear velocity, `vi` is the initial linear velocity, `a` is the linear acceleration, and `t` is the time.
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