How to Calculate Time for Particle Movement in 1D with Potential Energy?

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SUMMARY

The discussion focuses on calculating the time for a particle to move in one dimension under the influence of a potential energy function U(x) = c/x. The integral for time, t = ∫1/√{2(E - U(x))/m} from x0 to 5x0, requires substituting the total energy E = mv²/2 - U(x). Participants emphasize the challenge of integrating with the velocity v present in the equation, leading to confusion about whether to express v in terms of other variables. The consensus is that a substitution for velocity is necessary to simplify the integral.

PREREQUISITES
  • Understanding of classical mechanics, specifically energy conservation.
  • Familiarity with integral calculus and techniques for solving definite integrals.
  • Knowledge of potential energy functions and their implications in motion.
  • Ability to manipulate algebraic expressions involving variables and constants.
NEXT STEPS
  • Study the method of substitution in integrals to simplify complex expressions.
  • Learn about the relationship between kinetic and potential energy in one-dimensional motion.
  • Explore examples of solving integrals involving variable limits and potential energy functions.
  • Investigate numerical methods for evaluating integrals when analytical solutions are complex.
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Students in physics or engineering courses, educators teaching classical mechanics, and anyone interested in the mathematical modeling of particle dynamics in potential fields.

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Homework Statement



Equation for a particle moving in one dimension with potential energy U(x):

t=[tex]\int[/tex]1/[tex]\sqrt{2(E-U(x))/m}[/tex] , integrated from x0 to 5x0

Suppose U(x)=c/x for x>0. Calculate the time for the particle to move from x0 to 5x0, starting from rest at x0. Leave the answer in terms of m,c,x0

Homework Equations



E=mv2/2-U(x)

U(x)=c/x for x>0

The Attempt at a Solution



I've been working on this for a few hours, subbing E and U(x) out will give an integral w/ a denominator of:

[tex]\sqrt{2(mv<sup>2</sup>/2-2c/x)/m}[/tex]

however that now puts a v into the mix and it specificaly says "solve in terms of m,c, x0". Should i be replacing v with an equation for velovity or somthing? this integral is getting way nasty.
 
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hey, I have this exact same problem, how does one do this?
 

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