Ask Me Help Desk - Substituting A function into a velocity equation

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Substituting A function into a velocity equation

So I am doing some theoretical work with rockets and had a few question regarding acceleration and air resistance.

Vf ² = Vo ² + 2(A)(T)
Where Vf ² is the final velocity, Vo ² is the initial velocity, A is acceleration, and T is time.

Instead of just substituting a constant for (A), can a function relating (A) with respect to (T) be substituted?

no. These equations assume acceleration is constant. Your equation is wrong.

The equation $V_f^2 - V_0^2 = 2ad$ comes from the following:

$ \displaystyle\int_{d_0}^{d_f} {a dx } = \int _{V_0}^{V_f} {v dv} $

If a is constant, this becomes
$ 2ad = V_f^2 - V_0^2 $

But if a is not constant, then the integral on the left hand side becomes more complicated:
$ 2 \displaystyle\int_{d_0}^{d_f} {a(x) dx } = V_f^2 - V_0^2 $

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