Solution

We'll solve this in the same way as the pendulum example, using conservation of mechanical energy. Here there are two forces acting on the sliding object, the force of gravity and the normal force ${\bf N}$, but here again, the normal force is acting perpendicular to the direction of motion, and hence does no work, so we only need to consider the force of gravity to get the work. It is a conservative force, so we can use conservation of energy.

We'll first talk about the initial state of the slide, and then its final state.

initial time:

The initial y value of the object is $h$ and its initial kinetic energy is zero. Therefore the initial energy is $E ~=~K+U ~=~ mgh$.

final time:

The final value of y is zero and its final speed we'll call $v$. So the final energy is ${1\over 2}mv^2$.

So now we equate the initial and final energies, since energy is constant. We get

$$v ~=~ \sqrt{2gh}$$ (1.20)

One thing you should think about is: how would you solve this problem using Newton's laws? In principle it's possible because conservation of energy is a consequence of Newton's laws, however it is by no means obvious how to do it. Note that the final answer is independent of the shape of the path. Is that really true?