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Problem 90 Hard Difficulty

A girl is sledding down a slope that is inclined at $30.0^{\circ}$ with respect to the horizontal. The wind is aiding the motion by providing a steady force of 105 $\mathrm{N}$ that is parallel to the motion of the sled. The combined mass of the girl and the sled is 65.0 $\mathrm{kg}$ , and the coefficient of kinetic friction between the snow and the runners of the sled is $0.150 .$ How much time is required for the sled to travel down a $175-\mathrm{m}$ slope, starting from rest?


8.17 $\mathrm{s}$


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Video Transcript

to solve this question, you have to calculate twenties acceleration off this lad, so we will choose the following reference frame. These will be the why access it makes 90 degrees with the slope, and this we will be my ex. Access. It's parallel the slope. Then we have to find what is the acceleration off this land in the axe direction. To do that, we have to use Newton's second law. Then it goes as follows, applying Newton's second law to the X axis. We have the following the Net force. Next direction is a close to the mask off the girl times her acceleration. Then we note that the net force in the X direction is composed by 123 forces. Three forces because we can decompose the weight into components, one that points in the Y direction, which is the weight component. Why and the norther components that points to the X direction, which is the weight component. X. Then we have three forces composing the net force in X direction. We have the wind force pointing to the positive X direction we have the weight force component X outside, pointing to the positive X direction and we have the frictional force which points the negative X direction. And this is the coast of the mass off the girl times her acceleration. Then we went to calculate her acceleration. So acceleration is you close to the wind force plus the weight forced, minus defection all force divided by her mass. We already know what is the value off the wing force. We also already knew what is her weight? We know what is her weight because we know that her mass isn't close to 65 kilograms. Every also know that the acceleration of gravity near the surface off the earth is approximately nine point Eric meters per second squared. Then her weight forced is equals to 65 times 9.8 and then we only have two. They compose the weight force and we can do that by using this 30 degree angle. So this is also already known Alternative. You need some work it's already known. Then we need to calculate what is the frictional force? This is the only thing that is left for us to discover. Now the frictional force and the situation is the connected frictional force. Because she's already moving. Then in that situation, the frictional force is given by this kinetic frictional coefficient times the normal force. And then we already knew what is the kinetic frictional coefficient. So we have to discover what is the value off the normal force. For that, we have to use Newton's second law for the UAE access on the Y axis we have that the net force the Y direction is the cost of the mass times acceleration on the Y direction. I noticed that this lead is moving in the direction but it's not moving in. This already is direction. Therefore, the why acceleration is a cost to zero. Meaning that the net force in the Y direction is out. Sir. Vehicles to zero. Then I noticed that the net force in the Y direction is composed by two forces the normal force on the weight force. Why component to the normal force which points to the positive Y direction minus the wait. Why component which points to the negative wider action is it goes to zero. So the normal forced is he goes to the weight force. Why component? Then we can plugging the value often normal force here calculates the fictional forced and placed these in his equation and calculate acceleration. So now let us calculate what is their value off everything. So if the acceleration will be given by we enforced, which is 105 new terms, plus don't wait component White Let us calculate what is the white component off the weight force that we're doing here? The weight force is equals to 65 kilograms. Times 9.8 There is the full weight force. How can we calculate what he's? The white and ex components noticed that we can extend the weight force down here to get direct on the triangle. So did he is a 90 degree angle, and this is a 30 degree angle. So the only possibility for these angle is that it is because to 60 degrees, because then 60 plus 30 is 90 and 90 plus 90 is 180 as it should be for a triangle. So this angle up here is an angle off 60 degrees. No, remember that the angle between the Y component and this slope is 90 degrees to therefore the 60 degree angle plus, these are now angle is it close to 90 before this angle must be 30 degrees, then we already know what is the value off this angle murder. To calculate the X and Y components, we can use the sign and the co sign off 30 degrees. So the sign off 30 degrees is the costume in this context here, the opposite side of the triangle, which is the way to component acts divided by they have Parker News, which is the weight then the weight component X is equal to the weight times This sign off 30 degrees similarly, for the white component used the co sign. The curse signs equals to the address inside off the triangle. So wait component Why divided by the high party news, this means that the Y component off the weight force is a cost to W times. Their co sign off 30 degrees. Then that is companies. American values. The fool Late force is equals to 600 and 37 new terms them The X component off the weight force is 637 times 1/2, which is the sign off 30 race and this is the cost to 318.5 new terms. The Y component is given by 637 times the co sign off 30 degrees and this is approximately 551 0.658 new terms. Now we have in America values for both components now then let this country any calculating the acceleration. So it's 105 plus the X component off the weight force. 318.5 minus defection will force the frictional force is it goes to the kinetic frictional coefficient which is close to 0.15 times the normal force which is the cause of the Y component off the weight force. Then we have 551 0.658 and these wolfing is divided by the mass off 65 kilograms. Thes results in an acceleration off approximately five 0.2 for two meters per second squared. Now we know the acceleration. So then we can complete what time it takes to travel. 175 meters for that. Let me organize my ward. Okay, Now remember that the position as a function off time is given by the neutral position plus the initial velocity times think divorce time plus the acceleration times the time interval squared divided by true in the situation, I can do one thing which is to choose her initial position to be close to zero. Her initial velocity is close to zero truth then the position as a function off time is given by the acceleration times the interval of time squared, divided by Chu. Then we have to plug in the value off the displacement divided you off acceleration also for the time. By doing that, we have the following 175 is equals to five points. True for true times T squared, divided by truth. Then we sent this term here multiplying in these terms dividing to get 175 times true divided by 5.2 for two is a question T squared then tea is equals to the square root off 175 times two divided by five point chew for truth and these results in approximately 8.17 seconds