How to derive x(t) equation from energy

[cream3.14159]

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)\}}}}<br /> \ = 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.

 

[tiny-tim]

welcome to pf!

hi cream3.14159! welcome to pf!

(try using the X² icon just above the Reply box )

… 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 ​

 

[cream3.14159]

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.