Graphs of Position & Kinetic Energy vs. Time: Solving U(x)=7J

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SUMMARY

The discussion centers on graphing position and kinetic energy versus time for a particle governed by the potential energy function U(x) = 4x², with a total mechanical energy of 7 J. The position graph is identified as a sine curve due to the oscillatory motion of the particle, while the kinetic energy graph is described as a cos² function. The relationship between potential energy and kinetic energy is established, highlighting that when potential energy is at its maximum, kinetic energy is zero. The mathematical reasoning for using cos² instead of cos(2x) is emphasized as essential for understanding the energy dynamics.

PREREQUISITES
  • Understanding of potential energy functions, specifically U(x) = 4x²
  • Knowledge of total mechanical energy concepts in oscillatory motion
  • Familiarity with graphing trigonometric functions, particularly sine and cosine
  • Basic principles of energy conservation in physics
NEXT STEPS
  • Study the derivation of the relationship between potential energy and kinetic energy in oscillatory systems
  • Learn how to graph sine and cosine functions in relation to energy dynamics
  • Explore the mathematical justification for using cos² in energy graphs
  • Investigate the principles of harmonic motion and its equations of motion
USEFUL FOR

Students in physics, particularly those studying mechanics and energy conservation, as well as educators seeking to clarify concepts of oscillatory motion and energy relationships.

dmayers94
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Homework Statement


A potential energy function is given by U(x) = 4x^2 and I found the total energy to be 7 J. They ask: Graph position vs. time and kinetic energy vs. time using this information and assume that the particle has an initial position of 0.

The Attempt at a Solution


I got the first part correct. It is just a sine curve because the object oscillates and is bounded by the total mechanical energy of 7 J. The second graph, KE vs. time, is a cos^2 function, but I'm not sure why. I know that as the potential energy increases, position increases, and when potential energy is at a max, KE is 0. The shape makes sense to me when I compare it to the position graph, but how did they come up with cos^2 mathematically? Why not cos(2x) instead?
 
Physics news on Phys.org
You need to work back from the main principles, to get the reason for using a cos^2 function. What is the main principle between energy, potential energy and kinetic energy? And what is the equation for this relationship?
 

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