A 1500 kg car traveling at 10 m/s suddenly runs out of gas while approaching the valley shown in FIGURE EX10.11. The alert driver immediately puts the car in neutral so that it will roll. What will be the car's speed as it coasts into the gas station on the other side of the valley? Ignore rolling friction.

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Hey, everyone in this problem, we have a skateboarder with a mass of 70 kg skating at a speed of 7 m per second when he approaches a curve road. And we're showing this in a figure the skateboard moves freely without any special effort. And we're told to neglect rolling friction. And we're asked to determine the skateboarder speed when he reaches the top of the road on the other side. OK. So looking at our diagram, our skateboarder starts at a height of 3 m and a speed of 7 m per second, gonna follow this curved road and he's gonna be on the other side of the road at the top. That's the right hand side in our diagram at a height of 5 m. And we're looking for the speed there and we're gonna call that speed VF All right. Now, we're given four answer choices in this problem. Option, a 3.1 m per second. Option B 4.9 m per second, option C 9.8 m per second and option D 2.1 m per second. So in this problem, we have some height and skateboarders at different heights of the road, we have speed and speed is something we're looking for. So we can use the conservation of mechanical energy. OK. So recall, but the mechanical energy is going to be conserved in this case. OK. So the initial mechanical energy emac knot is gonna be equal to the final mechanical energy emac f now recall that our mechanical energy is made up of a couple of different things. So we have the kinetic energy and then we have the potential energy. OK? And the potential energy has two components. We have potential energy from a spring and we have potential energy due to gravity. And so what we get is that knot force us knot os UG is equal to KF us us F plus UG fa we're using not an F to indicate the initial and final time points respectively. K is going to be our kinetic energy us is our spring potential energy and UG is our gravitational potential energy. Now, in this case, we have no springs, nothing going on with springs. And so both us terms are going to be zero and we have the other four terms to worry about. Now, kinetic energy recall is given by one half MV squared, the gravitational potential energy is given by MGH. So if we write that all out, we get one half M V not squared plus MGH nought is going to be equal to one half MVF squared plus MGHF. OK. And you'll notice that, that VF value that we're looking for is in this equation. OK. So that's great. Now, what about the other terms? Do we have everything we need? OK. Well, the mass we have, OK. But one thing you'll notice is that the mass term M is in every single term that we have. So we can actually divide by the mass all the way through in this equation gives us one less thing to calculate. Um So we can divide up that mass. We have V not squared that initial speed. Yeah, we know G the acceleration due to gravity, we know the initial height and the final height we're given that in a diagram. So we have everything we need. So let's go ahead substitute in our values and see where that takes us. So we get one half multiplied by V not squared. So 7 m per second, all squared plus G 9.8 m per second squared multiplied by H knot. That initial height of 3 m. This is gonna be equal to one half VF squared plus G again 9.8 m per second squared multiplied by HF the final height which is 5 m. All right. So let's go ahead and simplify what we can on the left hand side, we just have numbers. And if we work all of this out, we're gonna get 53.9. And our unit is going to be meter square per second squared on the right hand side, we still have our one half VF squared. And we're gonna want to solve for VF eventually we're getting there. And then this last term is gonna give us plus 49 m squared per second squared. All right, great. So now we wanna isolate for VF, we're gonna subtract 49 m squared per second squared from both sides. And then we're just gonna switch the sides around so that our VF terms on the left, it's just a little bit nicer to look at. So we have one half VF squared, it is equal to 4.9 m squared per second squared. Give yourself some more room multiplying both sides by two. We have VF squared is equal to 9.8 m squared per second squared and finally taking the square root, OK. The square root is gonna give us both the positive and the negative route. Don't forget that. However, in this case, we're asked just for the speed. So we only care about the magnitude we're gonna go ahead and take that positive route. And so when we take this square route, we're gonna get 3.13 m per second approximately. And that is at final speed of the skateboarder when they reach the top on the other side of the road, on the right hand side of our figure in comparing this to our answer choices, we can see that option A is correct 3.1 m per second. Thanks everyone for watching. I hope this video helped see you in the next one.

Verified Solution

Key Concepts

Conservation of Energy

Kinetic Energy

Potential Energy