# How to derive x(t) equation from energy

1. Using V(x)= -max, in the following equation:
$$\int_{x_0}^x \frac{dx}{\pm \sqrt{{\frac{2}{m}\{E-V\left( x\right)\}}}} \$$ = t - t0

to get:
x = x0 + v0 + at2/2

E is total energy and V(x) is potential energy. I have tried hard integrating it in various ways but do not seem to get the required result.

I would really appreciate in help or tips in this regard.

When I use E - 0.5mv^2= V(x), the denominator becomes v and really does not help at all. If I do not do that, and use V(x) = -max that does not help either. I do not seem to be reaching the required equation in any way.

Last edited:

tiny-tim
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hi cream3.14159! welcome to pf!

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… to get:
x = x0 + v0 + at2/2

When I use E - 0.5mv^2= V(x)

(it should of course be x = x0 + v0t + at2/2)

why are you using E - 0.5mv2 ?

this is a perfectly ordinary integral of (constant - 2ax)-1/2

show us what you get

Hi!

Thank you for the response. I solved it with someone's help. The mistake I was doing was to use 0.5mv2-m*a*x to replace E. However, using v0 and x0 instead of v and x in this expression works to give the desired result and also, one has to put t0 = 0.