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Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass 7.50 kg, is sliding to the left at 5.00 m/s, while the other, of mass 5.75 kg, is slipping to the right at 6.00 m/s. They hold fast to each other after they collide. (a) Find the magnitude and direction of the velocity of these free-spirited otters right after they collide. (b) How much mechanical energy dissipates during this play?

A. $-0.226 \mathrm{m} / \mathrm{s}$

B. $-197 \mathrm{J} .197 \mathrm{J}$ of mechanical energy is dissipated.

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Simon Fraser University

Hope College

University of Sheffield

McMaster University

so welcome to a new problem this time. We have to, like, free spirited. So this is This is what we're gonna have before. So this is what we've given. So we have to free spirited archers. Okay? We have to free spirit and archers. Um, and they are moving towards each other. You know, we have the fast one, eh? As a mass that happens to be five point seven five kilograms. And and thie ardor is moving towards thie, right? With the last initial velocity move six meters per second. And then the second order is, uh, moving has mass off seven point five zero kilograms and is moving towards the left with an initial blast e off negative because, you know, the left is negative, and then the right is is positive. So if you move in, tow us, right, that's positive. Movement was left. That's negative. So this is the second ardor has as of last e, that happens to be, um, so five negative five point zero zero meters per second. And then what's gonna happen is afterwards they'LL collide with each other. The two free spirited orders will collide with each other. Um, and So there's there's going to be a final final Kamen velocity for them, you know, Final Kamen velocity for them. So we we want to find out. I want to find out the magnitude and direction. Okay? Want the magnitude and direction off this final last year looking for these two organizations, and that's magnitude and direction. That's what happens after. And then the second thing that's in part ain't but the The second thing we want to find out is thie. Oh, the mechanical energy and lost in the process or what they call dissipated in the process firmed. In this case, you know, we have a frictionless surface, so we'll use the law of conservation off energy to show that thie initial momentum, it calls to the final momentum that will help us isolate the final velocity. So the initial momentum equals the final momentum. We can switch that, and when they collide with each other, then you they end up having a common velocity as part of the final momentum. The initial momentum off he off the R E is B a I. And then the final momentum of the second order is em be V b I Okay, baby, I I think we do have to change this to B I just to make sure we have the same naming convention that's consistent. And so, the final for this one, the final velocity happens to be a V eight. That's the initial momentum off the first order and then this the the initial momentum of the second order Come on, divided by the mass of the first order and then the the mass of the second one. Ah, this will be a minus. So we have five points seven five kilograms times positive six meters per second vine iss, because this is a minus muscle. The second one seven point five kilograms times that five meters per second already dealt to the minus. And you want to divide off that by the sum of the two masses off the two artists. So five point seven five kilograms plus seven point five zero kilograms. And that uses, um, final, final, final solution off on negative. So this is your final velocity becomes a killing too negative to point on a rather negative zero zero zero zero two, zero point two to six four meters per second in terms of the requirements of the problem. We'LL simply going to say that the no, the magnitude is just the speed, which happens to be zero point two to six fourth. It is second amended direction. It's towards the left because so seems that the heavier one will push The lighter want was was left Second Part of the problem requires us to determine the changes in mechanical energy or the mechanical energy loss, which is equivalent to the initial kinetic energy off the system. Okay, initial Connecticut and you minus the final kinetic energy of the initial kinetic energies is the Connecticut and be off. Um, the fast part of which is Amy. The initial squared plus second order, which is him B B initial squared on DH. Then we wantto subtract of the final final kinetic energy, which is the common Connecticut. And you saw one half mm eight plus m b the final squared because the final squared when you plug in the numbers. But this problem you get one, huh? M is massive. Hey, if you go back, you see Marcel Hayes. Five point seven five kilograms killed them. And then the initial velocity of happens to be six meters per second. I want to square that plus one half must be happens to be seven point five zero meters per second. Okay, seven point five zero, not leaders second but kilograms the times the velocity of B which is negative five meters per second. The initial blast which is negative five point zero zero. Well, second, the sign of the blast is is make a difference because you wanna square it anyways and then one half mass off a's five point seven five kilograms plus mass of B seven point five zero kilograms and then you wantto puttin the final velocity Write that for you squared We got the final blast ia's point two two six for it's negative. They give zero point two to six four meters per second and that's going to be squared. So if you plug in the numbers for that if you plug in the numbers uh, you know, seven five negative five squared. Plus that the energy remember energy dissipated is going to be in terms of jewels and so and you plug in the numbers For this one, you get one ninety seven point two five jewels and you subtract the common energy between the two of them, which is, you know, round about zero point three three nine five jewels and therefore the energy lost of dissipated in the process. Um, you know, the mechanical energy becomes one ninety six one ninety six point nine jobs. Hope you enjoyed the problem. Feel free to ask any questions. Have a wonderful day and looking forward to the next problem. Okay, Thanks. Bye.

California State Polytechnic University, Pomona