# Conservation of energy quick question

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?

• Not that there's anything wrong with it, but uppercase $U$ is an unusual choice for speed. It usually means potential energy. Mar 23 '14 at 19:43
• @DavidZ it was phrased as such in the question, could you explain why it's not wrong?
– John
Mar 23 '14 at 19:53