How to derive x(t) equation from energy

In summary, the conversation discusses the use of the equation x = x0 + v0t + at2/2 to solve for the position of an object with constant acceleration. The use of E - 0.5mv^2= V(x) in the equation leads to a wrong result, but replacing v and x with v0 and x0 in the expression 0.5mv2-m*a*x gives the desired result when t0 is set to 0.
  • #1
cream3.14159
3
0
1. Using V(x)= -max, in the following equation:
[tex] \int_{x_0}^x \frac{dx}{\pm \sqrt{{\frac{2}{m}\{E-V\left( x\right)\}}}}
\ [/tex] = 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.
 
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  • #2
welcome to pf!

hi cream3.14159! welcome to pf! :smile:

(try using the X2 icon just above the Reply box :wink:)
cream3.14159 said:
… 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 ? :confused:

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

show us what you get :smile:
 
  • #3
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.
 

What is the relationship between energy and x(t) equation?

The relationship between energy and x(t) equation is that energy can be used to derive the equation for the position of an object over time, also known as x(t). This is because energy is a fundamental quantity in physics, and it is related to the motion of objects through principles like conservation of energy.

How do I derive the x(t) equation from energy?

To derive the x(t) equation from energy, you will need to use the conservation of energy principle. This principle states that the total energy of a system must remain constant, and it can only be converted from one form to another. By setting the initial and final energies of a system equal to each other and solving for x(t), you can derive the equation for the position of an object over time.

What are the units of the x(t) equation?

The units of the x(t) equation will depend on the units of energy used in the derivation. If the energy is in joules (J), then the units of x(t) will be in meters (m). If the energy is in electron volts (eV), then the units of x(t) will be in angstroms (Å).

Can the x(t) equation be derived from other principles besides energy conservation?

Yes, the x(t) equation can also be derived from other principles such as Newton's laws of motion or the work-energy theorem. These principles can also be used to solve for the position of an object over time by setting the initial and final conditions equal to each other and solving for x(t).

Are there any limitations to deriving the x(t) equation from energy?

There are some limitations to deriving the x(t) equation from energy. One limitation is that it assumes the system is in a closed and isolated environment where no external forces are acting on the object. It also assumes that the energy is conserved throughout the motion of the object. These assumptions may not hold true in real-world situations, which can affect the accuracy of the derived x(t) equation.

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