ShmoopTube

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Thank you We sneak and here's your shmoop du jour

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brought to you by amusement parks where we go to

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stuff our faces with junk food Then you get on

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the rides that are basically designed to induce motion sickness

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But we still have a blast And amusement park has

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a roller coaster with a forty five meter hill followed

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by a circular loop with a fifteen meter radius Assume

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that there's no friction on the track and that a

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roller coaster cars velocity is zero when it starts to

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descend the first hill Well which pair is closest to

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a car Speeds at the bottom and the top of

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the loop respectively Alright here the potential answers and meters

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per second Got it Okay well looking at this question

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we see that it involved potential energy and kinetic energy

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And well what does add up Tio that's right Funnel

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cakes No wait Sorry Anytime we're dealing with amusement parks

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are mined Just well Go straight to delicious fried dough

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Kinetic energy plus potential energy equals mechanical energy And since

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the total energy and a closed system can't change we

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can use this equation to figure out all sorts of

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stuff at the top of the track the coaster just

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sits there full of potential energy with no kinetic energy

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at all with kinetic energy it zero All we need

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to do is figure out the maximum potential energy and

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we can know the total energy in the system we'll

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potential energy equals mass times height times gravity so we

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can just plug the numbers in and well wait a

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second We don't know the mass of the roller coaster

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car I'll never mind folks Looks like we're stuck And

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this question is a knowable it's hopeless there's no possible

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way to well actually maybe we can figure it out

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after all Let's take another look at the situation We

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know what the top of the hill The coaster has

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all potential energy and no kinetic energy It stands to

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reason that if the bottom of the hill the coaster

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has no potential energy and all kinetic energy kinetic energy

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equals one half mass times the velocity squared But with

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this in mind we know that the maximum potential energy

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equals the maximum kinetic energy that means mass times gravity

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times height equals one half mass times velocity squared and

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with these equations balance like that mass cancels itself out

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Well we knew he could do it the whole time

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Using little algebra we can solve for velocity at the

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bottom of the hill That velocity equals the square root

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of two times gravity times height When we put in

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the numbers we find that the velocity at the bottom

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of the loop equals thirty meters per second Okay halfway

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there We probably could have ridden this roller coaster three

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times by now but it wouldn't be as much fun

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is solving this problem right And we've knocked out half

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of the answer's a and b you're definitely wrong So

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least there's some progress Now think about the energy at

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the top of the loop at this point With the

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roller coaster above ground level and moving around the loop

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we have both kinetic and potential energy going on the

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equation Looks like of this right here This thing in

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a potential energy equation the height is two times the

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radius of the loop since that gives us our total

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distance from the ground to the top Well this looks

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trickier Remember the total energy and the system can't change

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Knowing that we can use the same equation we used

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before but just factor in the change in height So

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the height in this equation will equal the starting height

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minus the height of the loop Right there When we

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put in the numbers we find the velocity at the

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top of loop is seventeen meters per second So our

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answer is d now considering our choices we really could

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have ruled out see as soon as we figured out

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the velocity at the bottom of loop that's because we

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know that the top of the loop is lower than

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the top of the hill and that means that the

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potential energy at the top of the loop has to

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be lower than it was at the top of the

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hill And if the potential energy of the system is

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lower than it was at the max then kinetic energy

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has to be greater than zero for the mechanical energy

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to be conserved There's no way The coasters velocity could

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be a zero Plus what kind of roller coaster stops

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at the top of the loop like no one would

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want to ride that thing All right well now we're

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going to go lay down All this physics and roller

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coaster talk has our head spinning We need some grandma

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me and about a pound of cotton candy Sounds good 00:04:08.995 --> [endTime] too Yeah

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