Work and Energy Legacy Problem #17 Guided Solution

In this problem we have a description of Nicholas who starts at the top of a racing slide, a water slide, and finishes at the bottom. In the midst of the problem, we are told that we can assume that in going from the top of the slide to the bottom of the slide that there are negligible losses of energy. Thus, the total amount of energy possessed by Nicholas at the top of the slide would equal the amount that he has at the bottom of the slide. We are given the mass of Nicholas, m equals 72.6 kilograms, and the height of Nicholas at the top of the slide, h equals 28.5 meters. We can presume that at the top of this racing slide, Nicholas is at rest, his v equals zero meters per second. The problem takes us through a stepwise solution to ultimately determining the speed of Nicholas at the bottom of the slide. In part A of this solution, we are asked to calculate the potential energy of Nicholas at the top of the slide. We presume this to be relative to the bottom or the ground. So we are going to use the equation p e equals m g h, and in using that equation, p e equals m g h, we plug 72.6 kilograms in for m, we plug 9.8 newtons per kilogram in for g, and we plug in 28.5 meters in for h. When we do that, we get the potential energy of Nicholas at the top of the slide. It comes out to be 20,277 joules. I can round that to two significant digits, 2.03 times 10 to the fourth joules. In part B, we are asked to determine the kinetic energy of Nicholas at the top of the slide as mentioned. We presume he starts from rest. So his kinetic energy at the top of the slide is zero joules. In part C, we are told, assuming negligible losses of energy, what is his total mechanical energy as he arrives at the bottom of the slide? Well, if no energy is lost, we can say that the energy at the bottom would be the same as the total amount possessed at the top. The total amount at the top is the 20,277 joules of p e plus the zero joules of k e. This comes out to be 20,277 joules. In part D and E, we are asked to determine the potential energy and the kinetic energy of Nicholas at the bottom of the slide. Potential energy is quite straightforward. He is down at the bottom of the slide where the height is zero, so his potential energy is zero joules. That would mean that all of his energy is in the form of kinetic energy, and as such his kinetic energy at the bottom of the slide is 20,277 joules. The final part of the problem asks us to calculate the speed of Nicholas as he arrives here at the bottom of the slide. So what we will have to do is associate the kinetic energy with the speed using the equation k e equal one half m v squared. The k e is 20,277 joules. The m is 72.6 kilograms. So I will need to substitute those numbers in. And then to solve for v squared, I have to get it by itself on one side of the equation. I can do this by multiplying both sides of the equation by two and dividing through by 72.6. This would give me v squared equal 558.6. And now I can take the square root of that and I get 23.6347 meters per second. I can round that to the first decimal place.