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# Does an object's speed at the bottom of a frictionless ramp depend on the shape of the ramp or just on its height? Explain. What if the ramp is not frictionless?

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In this problem we consider two cases: frictionless and with friction. Assume that the initial speed is $0$ and the height of the ramp is $y$. Let $v$ be the speed at the bottom of the ramp. The equation for conservation of energy is

\begin{align*} K_i + U_i &= K_f + U_f \\ 0 + mgy &= \frac{1}{2}mv^2 + 0 \implies v = \sqrt{2gy} \end{align*}

The final expression for $v$ is only a function of the initial height $y$, hence the final velocity of an object sliding down a $\textit{frictionless}$ ramp does not depend on the shape of the ramp.

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