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I have given one-dimensional motion of the particle directed horizontally. A problem says: "...Show that for this given motion Conservation of Energy Law holds.".

Since Energy can intuitively be thought of an operation that changes something over time, energy being conserved essentially means that the motion is not a function of time, but position only?

I.e. does energy conservation mean that a process is time-independent?

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Kinetic Energy + Potential Energy = Constant –  jinawee Jul 3 '13 at 12:18
    
I know that, but how to show that it is constant? –  user23709 Jul 3 '13 at 12:24
    
What's the motion formula? –  jinawee Jul 3 '13 at 12:34
    
I will write text of a problem. Air puck is moving on the smooth horizontal table and it is fixed on one end of a elastic spring with a constant stiffness factor $k$ and natural lenght $a$. Friction can be ignored. Show that for this movement Conservation of Energy Law holds. By the Hooke's law, I have the force $F=-k(x-a)$. I got that potential energy is $V=\frac{1}{2}k(x-a)^2$. Kinetic energy is $m\dot r^2$ –  user23709 Jul 3 '13 at 12:45
    
I've added the homework tag. For homework problems, please use the homework tag. –  Ben Crowell Jul 3 '13 at 13:14

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It is true that by Noether's theorem energy is conserved when the action is depednent on position only and not time. However, I think they want you to write down the equation of motion (use $F=ma$) and then use that to show that the total energy is a constant. Write down the expression for total energy and take its derivative w.r.t. time.

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