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You and your friends are doing physics experiments on a frozen pond that serves as a frictionless, horizontal surface. Sam, with mass 80.0 kg, is given a push and slides eastward. Abigail, with mass 50.0 kg, is sent sliding northward. They collide, and after the collision Sam is moving at 37.0$^\circ$ north of east with a speed of 6.00 m/s and Abigail is moving at 23.0$^\circ$ south of east with a speed of 9.00 m/s. (a) What was the speed of each person before the collision? (b) By how much did the total kinetic energy of the two people decrease during the collision?

A. $2.26 \mathrm{m} / \mathrm{s}$

B. $-639 \mathrm{J}$

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University of Michigan - Ann Arbor

Numerade Educator

Simon Fraser University

Hope College

once again welcome to a new problem. This time we have to friends. So we have two friends on DA Um, they're on a frozen pond. So, you know, one of one of the friends is moving eastward. So if you think about east west, not south. So this is not south. This is east, and this is West s. So one of the friends is moving eastward. And that's, um uh, eighty kilograms. His name is some. So the mass off some. So this is some and some is moving eastward. The Marcel's some happens to be eighty kilograms. So this is the information that you've given. Uh, you're given that the moss off some is eighty kilograms and some is moving eastward this way. And then also, we have a big girl. Abigail is moving. Um, Abigail is moving northward this way. And well, given that the mass off Abby girl happens to be fifty kilograms, that's the information that we've given The gonna collide off the ah, you know, after they're gonna meet and collide. So both some and Abigail will meet and collide. So when they collide, some moves north east at an angle off thirty seven degrees you can sew. This angle will be thirty seven degrees and and then the other thing we have is Abigail moves south east this way, and she's going to move at an angle off twenty three degrees. That's what happens. We can extend that for purposes off, finding solutions to this problem and the other extra information that given other than the fact that these two angles So this is thirty seven degrees and this is also twenty three degrees were also given that the some is going to be moving with a final velocity. You know, the final off six made us the second, and an Abigail is going to be moving southeast at twenty three degrees with the final velocity. So this is this is final velocity of some. So, you know, we should change the labeling right there. So we'LL say, uh, you know, sums final velocity And this is a big girls final velocity. So I'll be girls finally lost his nine point zero zero meters per second and so we want to find input. A. So all that's the information that's given, So we want to find in part a of this speed before collision a speed before collision. That simply means that there was an initial velocity for some. And then there was also an initial velocity for, um, happy girl. So those are the two numbers. We want to find out the initial velocity of some on the initial velocity of the other girls. And then the other thing we want to find out this is in part B. And what do you want to find out the, uh, the decrease in Connecticut. Okay. We want to find out the decrease of the the decrease in kinetic energy in the total Connecticut Andy Uh uh. Off both of them. That's the information will given the fast thing we'LL do is think about the movements after collision in the X and the wind direction. So, in the ex direction we have of the the last e final velocity off some co sign thirty seven degrees. And then we also have final velocity off Abby girl co sign twenty three degrees. That's the in the ex direction in the white direction. Some is going upwards. Okay, so some is going up. What? We should have had this, you know, a little bit lost so you can get to see some of these variable resolutions. So that's, uh, some Some of this would be a final velocity off some sign off any servant that's in the white direction and then for Abigail, it's negative because of the South, his negative north, his positive East is positive. West is negative. So this is negative. Final velocity of Barbie girl Sign off twenty three. So that's what we have so gonna find Apply the law of conservation of momentum law off Conservation of momentum both in the X in the white direction both in the X and the wind direction. So we'LL say, um you know, full, uh, in the ex direction I will say the initial momentum in the X equals to the final momentum in the X the initial momentum in the ex who only have some So, um, ourselves some times the initial velocity of some ask the equal the the X direction the final ah, muss off some time's final velocity off some in the X and then plus moss off Abigail Time's final velocity ofthe other girl in the X Direction. So goes to find the initial lost your son and that just comes by dividing all these numbers. So you know final, we're going to do that in the next spade. So for purposes off, continuation of we get final velocity off some no initial velocity of summers. You, Khun C. I's going Toby Mass off sometimes final velocity of some in the ex direction causing thirty seven plus muscle. Abigail Time's final velocity of a bagel in the ah, in the ex direction. And then we're going to divide all of that by thiss value here the mars off some. So that's the information we were given eso When? When we plug in these numbers, massive sum is eighty kilograms. Times of the final velocity of some is six meters per second. Ah times co sign off thirty seven and then plus uh, masal a beagle fifty kilograms. Time's final velocity off baby girl is ah nine. Remember, we're given Faz nine meters per second, so this is nine meters per second bond. Then times co sign off twenty three degrees on. We have to divide all of that by the mass of some, which is eighty kilograms. So we play in the numbers just to make sure we're doing the right thing. So eighty times, six times co sign off the D Servant plus fifty times nine times we'LL sign of twenty three. Come on, then all of that divide by eighty and it gives this nine point nine seven. So that's the initial velocity of some happens to be nine point nine seven meters per second. Ah, that's what we find that on then the next big want to do for Abigail And now, using the conservation off momentum in the white direction, we're saying that the initial momentum in the Y equals to the final momentum in the UAE. Initial momentum in the y. Remember this time we have Abigail the wind direction going up. What? Some ass off, Abigail Times initial velocity of a big girl because to mass off some times, final velocity off some in the white direction. So sign off thirty seven. And then we also have a component for every girl. So mass of Abigail Time's final velocity of Abigail. This would be sign of twenty three, so t yet the initial velocity of a big girl, we have to take ah the final momentum in the white direction for some and then minus the final momentum in the white direction for Abigail, remember, the girl is going in the opposite direction, so ha momentum is minus negative. And then divide by the mass of Abigail. Gonna plug in the numbers. Massive sun is eighty kilograms, eighty kilograms times velocity of some six point zero zero meters per second. Sign off thirty seven a minus mass off my big girl is fifty kilograms. Um, the final of the last year's baby girl is six. Made us the second sign off twenty three all of ah, Marcel's a big girl happens to be, Ah, fifty kilograms. So when we plug in the numbers, we get that thie initial velocity of a big girl happens to be two point two six meters. Yes, I can. So that's that's what we have right there on. Then in the next page will look att, the kinetic energy. Um so you know, we'LL see the change in the change in Connecticut energy for some So we'Ll do the change in Connecticut energy for some And this one becomes, uh, dull telekinetic energy. Some is, um final kinetic energy for some so sums final Connecticut and you mind the sums initial kinetic energy. So this is one half em of the final, Uh, can I, the Kennedy for some minus squared minus of the initial kinetic energy for some initial. So we need to change this a little bit. No. So sums initial squared, and so we plug in. We have This is massive sums off eighty eighty kilograms. Time's final velocity of some is six meters per second squared minus. Initial velocity of some is nine point nine seven meters per second squared. And that gives us a negative energy changing kinetic energy. And then we're going to do the same thing for Abigail. So, you know, change change in kinetic energy for Abigail because to the final kinetic energy for Abigail, minus the initial kinetic energy for other girl, which he calls to one half em. Abigail V. Abigail Final squared minus the Abigail initial squared. And so we plug in the numbers masses fifteen point zero kilograms and then times the velocity is the final velocity for Abigail is nine meters per second squared on DH. Then we want a minus, you know? So just make sure this is this is different right here. so they don't collide. And then minus the initial velocity of Big Girl, which is two point two six made us the second we competed that in the previous apart on. Then when you simplify that, you get negative. Six hundred and thirty nine jewels. So we have to combine the of the change. The total changing kinetic energy becomes change. Sure. Some minus change for Abigail, which is negative. Twenty three twenty five, three, six Jules plus. Ah! Oh, you know what this this was This was supposed to be our big girls. This is a big girl. So you want to make sure that we have the right numbers right there? So this is one thousand eight hundred and ninety seven point three two jewels. And so that's the number we're going to put in right here. Eight one eight nine seven point three two jewels. Um, yeah, and that ends up giving us negative six thirty nine jewels. So we combine these two Ah, the change in Connecticut. For both of them, um, the total change in Connecticut so immense that the decrease the decrees in Connecticut aji is six hundred and thirty nine jewels like that hope you enjoy the problem. Ah, feel free to send any questions my way and have a wonderful day. Okay, bye.

California State Polytechnic University, Pomona