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California State Polytechnic University, Pomona

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Problem 98 Easy Difficulty

Jonathan and Jane are sitting in a sleigh that is at rest on frictionless ice. Jonathan's weight is 800 $\mathrm{N}$ , Jane's weight is 600 $\mathrm{N}$ , and that of the sleigh and immediately see a poisonous spider on the floor of the sleigh and immediately jump off. Jonathan jumps to the left with a velocity of 5.00 $\mathrm{m} / \mathrm{s}$ at $30.0^{\circ}$ above the horizontal (relative to the ice), and Jane jumps to the right at 7.00 $\mathrm{m} / \mathrm{s}$ at $36.9^{\circ}$ above the horizontal (relative to the ice). Calculate the sleigh's horizontal velocity (magnitude and direction) after they jump out.

Answer

$v_{C}=0.105 \mathrm{m} / \mathrm{s}$ to the right.

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Video Transcript

{'transcript': "once again we have a new problem. Thiss time. It's a sledge that's has two people inside. So we have. Ah, Jonathan. And then we also have his friend Jen. So the the mass, the weight ofthe Jonathan, you know the weight of Jonathan has given us is equivalent to eight hundred our Newtons. And so we can change that right away to the mass ofthe Jonathan Ah, which is eight hundred mutants off a nine point eight meters per second squared. And if you do the numbers for the problems, you get eight hundred divide by nine point eight. That's eighty one point six three kilograms on DH. Then also, we have Jen. So the moss, the weight off Jane is given us, um, six hundred Newtons. And we can change that to the mass of Jane, which happens to be six hundred Newtons off nine point eight meters per second squared and that's going to give us. We do the number six hundred divide by nine point eight. That's prevalent to sixty one point two two kilograms. So those are their masses and also the sled itself, the mass off the sledge, or rather, the weight of the sledge happens to be a thousand mutants, and so we can get the moss off the sledge by taking a thousand mutants defied by nine point eight meters per second squared. And that's going to give this a masal. Um, so this is a thousand divide by nine point eight. That's one hundred and two point zero four kilograms. Also, we're told you know, this this sledge is writing itself to rest. So initially. Ah, initially, the sledge is at rest. And then what happens is that they see a poisonous spider. And Jonathan who happens to be no on the left. So Jonathan jumps towards the left with a velocity off five meters per second. So this is Jonathan jumping towards the left, Uh, with the velocity off five meters per second and the angle is thirty degrees, and then we have on the other side. We have Joanne who jumps to the right with a velocity off. What? This is Jonathan. So this is huh? Yes. Joann jumps to the right or rather not joined. Jame jumps to the right with a velocity of seven point zero zero minutes. Second button angle off. Thirty six point till this angle. That is six point nine degrees. So this is the information that you've given. And we want to find the horizontal horizontal velocity off the sleigh in terms of magnitude and direction. Ah, after the jump, you know? So what's gonna happen once they jump? And we're going to do that on the next page? So was saying we have all this information that we've given so fast. Of all we think off Jonathan. No, You know, he is going that way. And so he he has a velocity off. Jonathan happens to be if you go back, he was five meters per second, five meters per second. And this velocity can be resolved into Jonathan's ex component and Jonathan's. Why component? But in the ex component is going in the negative direction. So this is negative. And therefore the the velocity of Jonathan in the X is given us, uh, negative. Regular velocity of Jonathan times call Sign off a data. This state I studied agrees so, you know, watch elite study degrees to the horizontal. So we have to. But we have to remember that. So going back on this one, we should change these numbers a little bit. So this one is thirty degrees to the horizontal, and this one is thirty nine degrees also to the horizontal. So we have to be careful with those angles. So this is this one happens to be, um this is thirty degrees, and this is that is six point nine. We're so going back, we get the velocity is five point zero zero meters per second and then we want to multiply that by co sign off already. And so this gives us negative four point three three meters per second. Then we do the same thing for John. We say going Ah, you know that when it was the right, this angle is thirty six point nine degrees. So on the joins the last or Jen's velocity, I don't know why I'm saying joined all the time. James Velocity happens to be seven point zero zero meters per second. So her horizontal velocity, this's Jen's velocity in the X, and this is Jen's velocity in the Why and the X gens velocity will be a regular The last times call sign off data. So this is positive seven point zero zero meters per second times co sign off. That is six point nine degrees. And this is going to give us a five point six zero meters per second. So the these are they've lost is in the X on the next page will use the law of conservation of Momentum. Okay, the law ofthe conservation off momentum. Remember, once the jump off, um, you know, Jonathan has a momentum. The slant hasn't momentum. And also, Jane has a moment. So this is what happens once they jump off a moving that we're moving that way. So we want to find ah, what direction? What will will the sledge me be moving? So in this case, we say mass of Jonathan Times velocity of Jonathan in the X Plus mass off the sledge times velocity of the sledge. We don't know what the velocity of the Saudis, Time's, Marcel's, Jen and Velocity of Jenny in the X. That's gonna be zero in terms ofthe momentum because ofthe conservation. So solving way end up with M s V s equals to move these two to the right. Both of these go to the right. So we have I can't muss off Jonathan. The last year, Jonathan in the ex becomes negative, minus Marcel Jane of the last year of genuine. The ex also becomes negative. Then we have to divide by the mass off the sledge and both sides. And so the velocity of the sled becomes negative. You have to go back and get the mass of Jonathan. Eighty one point six three kilograms times the last day of Jonathan in the X, which is negative for point three three meters per second. Then minus Jen sixty one point two two kilograms and velocity of Jenny in the ex, you concede happens to be five point six zero meters per second. Okay, um, divided by the weight or the moss off the sled, which happens to be this one right here. One hundred two point zero four, you know, and we had to do the conversions. You could have waited up until the end to do the conversions. So then we finally find out that the velocity off the sledge is positive. Gonna be positives. The one zero point one zero five meters per second to the right. So that's what you have in that problem. Hope you enjoyed it, Phil. Free to send any questions of comment on DH comments and make sure you have a wonderful day. Okay, Thanks. Bye."}

California State Polytechnic University, Pomona
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