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I have a plate at potential $-1000V$ . What initial velocity should an electron ( at infinity) have so that it just strikes the plate. I don't want to use $\frac{1}{2}mv^2 = eV$ here.

I want to form an equation using Newton's law. The force varies and so does the acceleration. Now I want to find a differential equation depicting this result. I know that I will have to form a differential equation because force will change at every point and $F = qE = q\frac{dv}{dx}$ and I will have to use initial values. I am a beginner in differential equation and i am not able to form one.

Please take any assumption that needs to be taken besides the facts given because my motive is learn how to form differential equations in such cases .

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    The thing is the differential equation will lead identically to the result for conservation of energy since $$ F = m a = m \ddot{x} = m \frac{d \dot{x}}{d t} = m \frac{d x}{ d t } \frac{d \dot{x}}{ d x} = m v \frac{dv}{ dx} = \frac{1}{2} m \frac{d}{dx} (v^2) = \frac{dT}{dx}$$ where T stands for the kinetic energy – 2017-02-27
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    @Triatticus Can you please explain it " the Sum of a series ". Like to calculate the integral of xdx we can d (x+ dx)dx + (x+2dx)dx + (x+3dx)dx....... and the sum it using arithmatic progression. Can you explain the above example in that sort of I way because I just could't move a step . Thanku – 2017-03-02
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    I can list the steps, but the nice thing about the method I mentioned is that it is independent of the force considered as long as it depends on position. I'm unsure what your comment about series means but I can explain the steps of solution better when I get to my computer – 2017-03-02
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    @Triatticus Thank you ! Yes I understand the above integrating process is the best . But I am saying that we are taught about integration being the limit of a sum. We use it to prove the integration of x , x^2( because they form geometrical or arithmetic series), sinx , cosx . Like this http://www3.ul.ie/~mlc/support/Loughborough%20website/chap15/15_1.pdf..Also I understand that all integrals can't be evaluated by summing its series because we don't have the methods to sum all the series possible. – 2017-03-02
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    @Triatticus But if you could just show me how the above integral can be expressed as terms of a serie just as in the above link it would very helpful . Thank You ! – 2017-03-02
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    That sounds like it should be an entirely different question and has little do do with your initial question – 2017-03-02
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    @Triatticus Ok. Can you please write an answer if you wish. I wanted to know this . It would be a great help. Thank you. – 2017-03-05

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