If the initial angular velocity is not 0, then we cannot simplify the formula as we have done before. Whereas previously, the formula can be simplified as follow:

ω = ω₀ + αt
ω = αt
α = ω / t

We now must account for the initial angular velocity:

ω = ω₀ + αt
ω - ω₀ = αt
α = (ω - ω₀) / t

For example, if the initial angular velocity is 50 degrees/s, we can find the angular acceleration required to accelerate to 250 degrees/s in 0.2 seconds like so:

α = (ω - ω₀) / t
α = (250 - 50) / 0.2
α = 200 / 0.2
α = 1000 degrees/s²

As a reality check, we expect the angular acceleration to be lower when the initial angular velocity is higher, all else being equal. This is because the difference between the initial and final angular velocity isn't as great, therefore we needn't accelerate as much to get to the same angular velocity. This is consistent with the result we got (1000 < 1250).

Finally, note that the term ω - ω₀ is the **difference** between the final angular velocity and the initial angular velocity, which can be rewritten:

Δω = ω - ω₀

And thus, we can rewrite the derived equation, α = (ω - ω₀) / t, as:

α = Δω / t
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