Work and Energy Legacy Problem #28 Guided Solution

A good problem solver reads a problem carefully, develops a mental picture of what's going on, identifies the known and the unknown quantity, and then plots out a strategy for getting from the known to the unknown using physics principles and math relationships. Here we read of a toboggan that's kicked on a frozen pond and given some kinetic energy. Thanks to the friction force that acts between the ice in the pond and the toboggan, the toboggan begins to slow down and eventually comes to a stop. We're given that m of the toboggan, 6.8 kgs, and the initial speed of the toboggan, 1.9 meters per second, and the mu value, the coefficient of friction, equal 0.13. We're asked to determine the distance that the toboggan slides before coming to a rest. The solution will center around the use of the five-term equation that you see listed here on this page. On the left side, we see there's a PEI, and on the right side, we see there's a PEI. Since this is on a level surface, the PEI and the PF are the same value, we could even call them zero, and as such, they'll cancel from both sides of the equation. We have some initial kinetic energy. It's one-half times 6.8 times 1.9 squared. We can calculate this amount, and we'll get 12.274 as the KEI. We have no kinetic energy, finally, on the right side of the equation, so the entire right side of the equation becomes zero. We could take this five-term equation now and simplify it to two terms, with the first term on the left side being KEI, or 12.274, and the second term being W and C. On the right side, of course, we have zero. So we can now find W and C. It's equal to negative 12.274 joules. Now this W and C is done by friction. It's work done by friction, and friction force value is not explicitly stated here, but if we were to draw a free body diagram for the toboggan, we'd have three forces acting upon it as it slows to a stop. There would be the downward force of gravity, whose value would be mg, the upward normal force, which would balance this downward force of gravity, and thus also would have the value of mg, and there would be the friction force opposing the motion of the toboggan. The friction force is always calculated as mu F norm, and here the F norm is mg. So the friction force is mu, the coefficient of friction, times m, the mass, times g, 9.8 newtons per kilogram. Substituting values of 0.13 for mu and 6.8 for m, we would get 8.6632 newtons as the force of friction. Now the work, negative 12.274, done by friction, is equal to the friction force times the unknown d quantity times the cosine of the angle between F and d, which is 180. Substituting in the value of F, we can solve for d, and we get 1.4168 meters. Rounding to two significant digits, that would be 1.4 meters.