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A roller coaster reaches the top of the steepest hill with aspeed of 6.0 $\mathrm{km} / \mathrm{h}$ . It then descends the hill, which is at anaverage angle of $45^{\circ}$ and is 45.0 $\mathrm{m}$ long. What will itsspeed be when it reaches the bottom? Assume $\mu_{\mathrm{k}}=0.12$ .

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$=23.49 \mathrm{~m} / \mathrm{s}=8 \mathrm{5km} / \mathrm{h}$

Physics 101 Mechanics

Chapter 5

Using Newton's Laws: Friction, Circular Motion, Drag Forces

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

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Rutgers, The State University of New Jersey

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So let's first convert. Ah, the speed from kilometres for ours to meet us for a second. And, uh, let's call the velocity as V zero So easier is six communists for our times. One use for a second, divided by 3.6 kilometers for our so that gives us 1.667 meter per second. Now the total displacement in the ex direction. So what's extraction? First of all, let's draw the free body diagram of the roller coaster. So we have the roller coaster which is going downhill. It's good. And ah, if we consider the downward direction parallel to the slope as positive X, then the perpendicular direction two x will be why and we're taking upward direction. It's positive. So the displacement is from the top of the, um, hill to the bottom. And if we call the top as X and bottom is ex knows, then the displacement is 45 meters. What if I find Tonto now? Um, we have a bunch of forces acting on the roller coaster. First of all, we have md which is acting downwards, then parallel to why we have the normal force. Then we have the friction, which is acting parent to expert in the opposite direction. Tow the motion of the roller coaster, which is f off fr. And finally, finally, um, we see that if the hill makes an angle data, then the then the component of mg along why direction will be mg course science data, because the angle between why and ah, vertical direction off mg is set up. And Caroline do X, it's gonna be MG science data. Okay, so now we can use ah Newton's second law in ex direction. So we see we already have already said the direction off X in directional fix. And that's downward direction is positive. So we have m d signs it on minus f f r which is sometimes a and ah, in our case, this it will be zero because, um, because we don't have any ax elation that's on the roller coaster. Sorry. Um, we do have acceleration, and that's that's what we're finding. We don't have an explosion. Why access? Um, so in why direction will have mg cool science data. Um, plus f off end. So this will be negative because we've set our projection is positive equals zero Or in other words, if off anything will do mg co Santa. So yeah, there's no taxation on why direction, but yeah, there's definitely exploration and extraction now. Ah, in extraction. If if we, uh, said the fiction equal toe kind of friction then we see that if off, if our must be equal to mu k times f off n where n is the normal force And we've already seen that effort physical the NGOs and leader So if we combine everything together, we see that e will become mg It's actually delighted in a separate cage, so a, um so f off X will be equal toe mg signs ada minus new K f n, which is equal to m g science data minus mu K and G co sign Rita. And from here it was all for a That's gonna be n g sign Veda minus mu s mg course and divided by and And we can get rid of the EMS. And finally we get tea pains Scientist on minus nuke a co sign. Now we can use the question two point well see to solve for defining velocity. So there's a question tells his V minus V zero squared sequel to a Times X minus x enough Now, in our case, we know all the factors we know exploration or we can know that from the previous calculation. This part. So we know a we know it and it's ex not and we also know be not so. We can easily find the squid on the way we find that is really light B squared equals to be not squared plus two a x minus x not. And from there, if we take the square root, we end up having this equation Now we substituted a here. We knew that we know that a single G signed ADA minus mu ke course and data and if we used the given numbers, we see that the knot is 1.667 meters per second. Leda is 45 degrees G's 9.8 meters per second squared and mucus Giving this 0.1 toe and also explain its X note is the displacement, which is 45 meters. So combining all of this together we see that the final velocity is 23.49 meter per second meters per second. And if we convert that kilometers for our it's gonna be 45 parameters for our thank you

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