Work and Energy Legacy Problem #9 Guided Solution

This is a difficult problem that will require that you follow the habits of an effective problem solver. Those habits involve reading the problem carefully, developing a mental picture of what's going on, identifying the known information in the unknown quantity, and then thinking about mathematical relationships and physics concepts in order to get from the known information to the unknown. Here we have a problem about a ski slope and a tow rope that's pulling skiers and snowboarders up the hill that's inclined at 14 degrees. We're told it's a constant speed motion, we're told that there's 18 skiers on the rope with an average mass of 48 kilograms, and we're told the motor is a 22 kilowatt motor. Fortunately for us, the problem steps us through the solution in somewhat of a stepwise fashion, asking first what the cumulative weight of all the skiers is. So we have 18 skiers, a mass of 48 kilograms on average, and the weight is simply m times g. So we need to take our 48 kilograms, multiply it by 9.8 newtons per kilogram, and finally multiply it by 18, since there's 18 skiers. That will get us a cumulative weight value of 8,467.2 newtons. I'll round that to two significant digits, such as 8.5 times 10 to the third newtons. The second question is going to ask me to determine the force which would be required to pull this amount of weight up the 14 degree incline at constant speed. Now this is information that I need to draw on from the previous unit. When we spoke of incline planes, we spoke about the force of gravity parallel to the incline plane is simply mg, calculated in part A, multiplied by the sine of the angle of incline, which here is 14 degrees. So to determine the force applied parallel to the hill and up to cause a constant speed motion, I need to find the parallel component of gravity. And that would be mg, or 8,467.2, multiplied by the sine of 14 degrees. When I do that, I get 2,048.40 newtons. That's the force applied parallel to the hill to cause a constant speed motion. Now in part C of the problem, I'm asked what the speed would be for a 22 kilowatt motor applying this force. Now I'm used to relating power to work and time, but here it's going to be difficult to calculate the work since I don't know the distance. So what I'll have to do is derive an equation that relates power to force and speed. The derivation goes something like this. Power is equal to work divided by time. And work is equal to F times D. So power is equal to F times D divided by T. Now there's a D and a divided by T in that mathematical statement. So power is equal to F times D divided by T. Well, recognize D divided by T to be a speed. So power is equal to F times V. And rearranged, we could say that V is equal to power over work. Now the power is 22 kilowatts, not a nice unit, so we need to convert that to 22,000 watts and divide that power in watts by the force in newtons of 2048.40 newtons. When we do, we get 10.74 meters per second, and we can convert that to two significant digits, 11 meters per second.