Work and Energy Legacy Problem #27 Guided Solution

A good problem solver reads the problem carefully and begins to develop a mental picture of what is going on, identifies the known information and the unknown quantity to be solved for, and then uses physics understanding and math relationships to plot out a strategy as to how to get from the known information to the unknown information. In this question we read of a roller coaster car which is traveling along a level section of track at a given initial speed of 22.8 meters per second. The car abruptly breaks to nearly a stop, to 2.9 meters per second, over a distance of 5.55 meters. We're told that m of the car is 328 kilograms and our unknown quantity is the braking force which acts upon the car. Now the solution to this problem is best centered around the five term equation that you see at the bottom of the page. In this equation we have the PEI on the left side and the PEF on the right side. These PEIs and PEFs are based upon height and this is a level section of track and as such the height initial is the same as the height final. Whatever the PE initial and PE final are, they are the same and since they are being added to each side, we can cancel them. Now we also have the KE initial on the left side and we can calculate KE initial is one half MVI squared, the M is 328 and the VI is 22.8. We can do the same thing for the KE final since we have a final velocity of 2.9 meters per second. The W and C term has within it an FD cosine theta where the F is the braking force that we're trying to solve for, the D is 5.55 meters and the theta is the angle between F and D since it's a braking force that angles 180 degrees. We wish to calculate the braking force and to do so we'll have to calculate KEI and KEF and then solve for W and C. Once we do, we'll set it equal to FD cosine theta and solve for F. Here's how we'll do it. KEI can be calculated here. If we go one half times 328 times 22.8 squared, we get 85,253.76 J. KEF can be calculated if we go one half times 328 times 2.9 squared and we get 1,379.24 J. Now if we find the KEI minus the KEF, what we're doing is we're finding the negative of the work value. So doing that gives us 83874.5 J and that value is equal to the negative sign of the work. So now we know what the work is. It's negative 83,874.5 J and that work quantity is equal to the braking force F times 5.55 times the cosine of 180 degrees. Solving for F, we get 15,112.5 Newtons. Rounded to three digits, we'd have 1.51 times 10 to the fourth Newtons.