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$\bullet$ $\bullet$ Ski jump ramp. You are designing a ski jump ramp forthe next Winter Olympics. You need to calculate the verticalheight $h$ from the starting gate to the bottom of the ramp. Theskiers push off hard with their ski poles at the start, just abovethe starting gate, so they typically have a speed of 2.0 $\mathrm{m} / \mathrm{s}$ asthey reach the gate. For safety, the skiers should have a speedof no more than 30.0 $\mathrm{m} / \mathrm{s}$ when they reach the bottom of theramp. You determine that for a 85.0 -kg skier with good form,friction and air resistance will do total work of magnitude4000 J on him during his run down the slope. What is the max-imum height $h$ for which the maximum safe speed will not beexceeded?

$h_{\max }=50.5 \mathrm{m}$

Physics 101 Mechanics

Chapter 7

Work and Energ

Physics Basics

Applying Newton's Laws

Kinetic Energy

Potential Energy

Energy Conservation

University of Washington

Simon Fraser University

University of Winnipeg

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so that the first draw the ramp and then we can talk about how to deal with the problem. So let's say this is the ramp and this is the final point and that's the initial point. So the ski, when the skill reaches at the initial point, he has a velocity off two meter per 2nd 2nd and a defining point. Here's a velocity of 30 meters per second, so the F is 30 meters per second and G I is toe, I mean for a second. And we know that during his slide there is some air resistance and ah, frictional force acting against the slight which so the total worked and due to that is 4,000 jewels. So we put a negative sign there and the reason I put in a great sign is because a dragon friction forces acting against his slight. So that's why I worked on his negative. Now what if we if we said this as our origin in the white direction, then we can say that why f or the final y zero and this why I is Ah, let's say the height and we call hide as H. So let's say this is the height so we can use the energy conservation here, which will be key initial plus K k initial Plus you initially plus worked on you too. Some other factors, for example. The friction and eject should be equal. Do he finally plus you find Now we know that there's no potential energy stored here because we have said that 0.0 so we can ignore this part. And what what happens to the final candidate energy. So we know that the mass is already given, which is 85 kg and the final velocities 30. So Kiev will be half off M V f squared. And if we use the numbers, we'll have 13 seconds squared will have the candidate energy as 38,000 2 50 Jules. All right, so this is the amount of energy that should be at this bottom point. So at this point, the total mechanical energy or and the addition off kinetic energy and potential, and it should be no more than 38,000 to 50 jewels, plus the other form off energy which wass the friction of energy, which gives us a 42 1,000 to 50 Jews So the way we got that is Web set K r a k I. Plus you, I minus 4,000 Jules equal to 38,000 to 50 jewels. And from there we got K II plus new eye as 42,000 to 50. Jules, Now we know that, kay. I will be half off m the I squared, which is half off 85 kg times to meet over second squared, which is 1 70 Jules, if you use that number, we see that u y should not be more than 42,000 to 50 Jules minus one seventeen 17 jewels which is for the 3,000 0 aids your jewels. And we know that since you have said this as a judge, so our potential energy you y is m times g times h and we sent that toe 42,000 08 Serial Jules. From here, we can get the hide us for 2,000 08 Serial Jules, divided by sometimes Chief, where m is 85 aunties 9.8. So using the numbers have 85 kg times 9.8 meters per second squared. It gives us a height off 50.5 meters. So that's the, uh, maximum allowable height that we need to build. Thank you

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