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Say we had a particle moving in a frictionless funnel and was projected horizontally.

If we had some initial conditions for the energy E, then would these conditions be the same always?

For instance, in this particular question I got $$ E = 1/2m\dot r^2 - mgz, $$ and we were given that $z = b\left ( \dfrac{b}{r} \right )^n$, and it was projected at the inner surface level $z = b$ horizontally with speed $U$. Using those initial conditions, I got $ E = 1/2m U^2 -mgb.$ However would I be correct in the stating that $$ 1/2m\dot r^2 - mgz = 1/2m U^2 -mgb?$$ I looked in the solutions and the lecturer wrote that they were equal, but does the energy not change of the particle?

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  • $\begingroup$ Not that there's anything wrong with it, but uppercase $U$ is an unusual choice for speed. It usually means potential energy. $\endgroup$
    – David Z
    Mar 23 '14 at 19:43
  • $\begingroup$ @DavidZ it was phrased as such in the question, could you explain why it's not wrong? $\endgroup$
    – John
    Mar 23 '14 at 19:53
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yes you can say that they energies are the same. And with energy we usually mean that the potential plus kinetic energy always remains constant. so kinetic energy can change and so does potential but the sum of those 2 is always constant unless you add friction.

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