EXAMPLE 5.8 Hit the Ski Slopes GOAL Combine conservation of mechanical energy with the work-energy theorem involving friction on a horizontal surface. Consider the following figure. h = 20.0 m х The skier slides down the slope and onto a level surface Customize and control Google Chrome. Update is distance d from the bottom of the hill. available. PROBLEM A skier starts from rest at the top of a frictionless incline of height 20.0 m, as in the figure. At the bottom of the incline, the skier encounters a horizontal surface where the coefficient of kinetic friction between skis and snow is 0.210. (a) Find the skier's speed at the bottom. (b) How far does the skier travel on the horizontal surface before coming to rest? Neglect air resistance. STRATEGY Going down the frictionless incline is physically no different than going down a frictionless slide and is handled the same way, using conservation of mechanical energy to find the speed v at the в bottom. On the flat, rough surface, use the work-energy theorem with W, =D nc -fd, where fis the W fric magnitude of the force of friction and d is the distance traveled on the horizontal surface before coming to rest. SOLUTION (A) Find the skier's speed at the bottom. Write down the equation for 2gh = V 2(9.80 m/s²)(20.0 m) = 19.8 m/s %3D conservation of energy, insert the values v O and yB = 0, solve for v B and substitute values for g and y, as the skier moves from the top, point O9, to the bottom, point ®. (B) Find the distance traveled on the horizontal, rough surface. mv? 글깨요 Apply the work-energy theorem as the W. net 2 - fd = AKE = -mv skier moves from B to ©. Substitute ve = 0 and -Hymgd = -mvg? 2 fk = Hkn = Hxmg. - fка = AKE = 2mvc net skier moves from B to © Substitute v, = 0 and 2 -Hymgd k = H = Hymg. Customize and control Google Chrome. Update is available. Solve for d. (19.8 m/s)2 2(0.210)(9.8 m/s²) d = = 95.2 m 249 LEARN MORE REMARKS Substituting the symbolic expression v, 2gh into the equation for the distance d shows that d is linearly proportional to h: Doubling the height doubles the distance traveled. QUESTION Give two reasons why skiers typically assume a crouching position when going down a slope. (Select all that apply.) O Crouching decreases the mass of the skier. Crouching lowers the skier's center of mass, making it easier to balance. Crouching decreases the skier's inertia. The acceleration of gravity g is increased by crouching. In the crouching position there is less air resistance. PRACTICE IT Use the worked example above to help you solve this problem. A skier starts from rest at the top of a frictionless incline of height 20.0 m, as shown in the figure. At the bottom of the incline, the skier encounters a horizontal surface where the coefficient of kinetic friction between skis and snow is 0.226. Neglect air resistance. (a) Find the skier's speed at the bottom. 19.79 m/s (b) How far does the skier travel on the horizontal surface before coming to rest? 88.50 EXERCISE HINTS: GETTING STARTED | I'M STUCK! Use the values from PRACTICE IT to help you work this exercise. Find the horizontal distance the skier travels before coming to rest if the incline also has a coefficient of kinetic friction equal to 0.226. Assume that 0 = 20.0°. | 4 61.02 What is the initial energy of the skier? How much work is done as the skier slides down the incline? How much work is done as the skier slides to a stop on the horizontal surface? In other words, write expressions for these three amounts of energy, then relate them to each other. m Need Heln? Read It