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Two arrows are fired horizontally with the same speed of 30.0 m/s. Each arrow has a mass of 0.100 kg. One is fired due east and the other due south. Find the magnitude and direction of the total momentum of this two-arrow system. Specify the direction with respect to due east.

45.0 $\mathrm{deg} \mathrm{SE}$

Physics 101 Mechanics

Chapter 7

Impulse and Momentum

Moment, Impulse, and Collisions

Rutgers, The State University of New Jersey

Hope College

McMaster University

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All right. In problem number six, we have two arrows. They're fired horizontally. One fired due east, the other fire due south. And we're being asked to find the magnitude and direction of the total momentum of this system. Yeah, so as a plan here, the first thing we're going to do is find the momentum that each arrow has. Then we're going to use those as the component portions of our triangle, and we're going to find the result in May momentum of the two aero system. Okay, so let's start with what we have. We know that each arrow has a mass Oh, 0.1 kilograms. So 0.100 kilograms and both are fired with the same velocity. I should say the same speed. Okay. Of 30.0 meters per 2nd 1 of these is fired east and one of them is fired south. No. First we need to find the magnitude of the mo mentum of these arrows. So let's start with their. So we'll momentum equals mass times velocity. So very simply, we can take 0.100 kilograms times their speed of 30 meters per second. And for each arrow we get a momentum of three kilogram meters per second. Okay, remember, that is for that's the magnitude of the momentum free chair. Oh, what we have going on is we actually have the arrows being fired in different directions so we can think about it. Kind of like this first arrow fired due east. Okay, we can call this arrow one. Then we have a row to being fired due south. I'm going to use the head tail method here to set up our triangle. And what we are trying to find is this resultant of the two aero system. And then the question in the book asks us to define it in terms of the angle from east. So we're gonna look for theta here. That's gonna be our direction. Now we know that momentum of the first arrow was three kilogram meters per second and the second arrow was also three kilogram meters per second. So we can choose Pythagorean theorem here to find the magnitude of the resultant. So this is our make our total magnitude. So three square plus three squared equals C squared. Which C would be the high pot news here and then When we solve for C, we end up with a magnitude of 4.24 kilogram meters per second. And remember, this is our momentum total. Okay, so that is the magnitude of our total momentum for the system. Now we need to find the direction, and to do that, we're going to actually use. I'm gonna use inverse tangent to find our theater that we defined in the problem. So in this case, Fada, well equal the inverse tangent of the rain kilogram meter per second. Bye bye. Three kilogram meters per second. Now, in this problem is in a great example of how to solve for this because the two sides are the same. But this top number is the opposite leg of the angle. And the bottom number is the adjacent leg of the angle of the right triangle. And we saw it through, and our theater equals 45 degrees. So if we were to write this out in a full final answer, we would say mo mentum total equals 4.24 kilogram meters per second and 45 degrees. And then we're going to call this so of East Star. Let's try that again to make that a little better for you. Okay. 45 degrees. This is south of East

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