Work and Energy Legacy Problem #16 Guided Solution

Here we have a problem about Susie Love to Ski, who is skiing on Bluebird Mountain. We're given the mass of Susie, m equals 56 kilograms, the speed of Susie, v equals 16 meters per second, and the height of Susie, h equals 34 meters. This is at the top or at the crest of a ski hill, and we're given this height information relative to the bottom of the run. Now we're asked several questions about Susie, which can be done sequentially. The first question asks us to calculate the kinetic energy. We understand kinetic energy to be one-half m times v squared. So to calculate the kinetic energy, I'm going to go one-half times the 56 times the 16 squared. And when I do, I get 7,168 joules, and I can round that to two significant digits, 7.2 times 10 to the third joules. In part B of the problem, we're asked to calculate Susie's potential energy, and that's relative to the height of zero at the end of the run. So finding this potential energy involves using the equation Pe equals mgh, where the m is 56, the g is 9.8 newtons per kilogram, and the h is 34 meters. Substituting into the equation and solving for Pe gives me 18,659. I can round this to two significant digits, 1.9 times 10 to the fourth joules. Now in part C of this problem, I'm asked to calculate the total mechanical energy at the same location. I understand that the total mechanical energy is simply the sum of the two individual forms of mechanical energy. So here I need to take the Ke and the Pe and add them together. I need to take the 7,168 joules for Ke and the 18,659.2 joules for Pe and add them together. Doing so gives me 25,827 joules. I can round this to two digits such that I get 2.6 times 10 to the fourth joules. Now in part D, I'm given the hint that no energy is lost or gained from the top of the hill to the arrival at the end of the run, and to determine the total mechanical energy at the end of the run. Doing this involves simply equating the amount of total mechanical energy at the top of the hill to the amount that she has at the end of the run. So the answer is pretty straightforward here. It's just 25,827 joules, which I can again round to two significant digits. Now in part E, we're asked to determine Susie's speed as she arrives at the end of the run and prior to breaking to a stop. What I know about Susie is I know she's at the end of the run where the height is zero meters and I know she possesses 25,827 joules of kinetic energy or mechanical energy. Not only is that the mechanical energy, that's also the kinetic energy since she has no potential energy at this location. So I equate this figure, 25,827 joules, to one half times the M of 56 times the V squared. And then I have to solve for V. That's going to require some good algebra skills. You'll need to multiply both sides of the equation by two and divide by 56 and take the square root of the result. You'll get 30.37 meters per second. You can round that to two digits such that the answer is 3.0 times 10 to the first meters per second.