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Problem 36 Easy Difficulty

To protect their young in the nest, peregrine falcons will fly into birds of prey (such as ravens) at high speed. In one such episode, a $600-\mathrm{g}$ falcon flying at 20.0 $\mathrm{m} / \mathrm{s}$ hit a $1.50-\mathrm{kg}$ raven flying at 9.0 $\mathrm{m} / \mathrm{s}$ . The falcon hit the raven at right angles to its original path and bounced back at 5.0 $\mathrm{m} / \mathrm{s}$ . (These figures were
estimated by the author as he watched this attack occur in northern New Mexico.) By what angle did the falcon change the raven's direction of motion? (b) What was the raven's speed right after the collision?


(a) $\theta=48.0^{\circ}$
(b) $v_{r 2}=13.45 \mathrm{m} / \mathrm{s}$


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Video Transcript

In this question, we have a collision between two birds, a paragraph falcon and a raven. The falcon is 600 g and it is flying at 20 m per second and hits a 1.5 kg rave in flying at 9 m per second, the falcon hits the raven at right angles and bounces back at 5 m per second. And where we are to calculate the angle that the Falcon is able to change the Ravens direction of motion by and additionally, the Ravens speed after the collision. So to get us started off, I'd like to dio a quick picture of the before scenario as well as the after so before the collision We've got the rave in flying along in this direction. Say, I'm gonna call this m r for the rave in and it is traveling at an initial speed. I'm going to call this V R I of 9 m per second and then we've got a falcon coming in at right angles to the to that path that the raven is executing and it is flying at a speed of 20 m per second. So we're gonna label that VF for V Falcon I for initial, and that's going to be 20 meters per second. So this is our initial condition. And then after the collision, presumably the Ravens course has been changed or altered a little bit. So this is the rave in, and we don't know what its final speed is. That's something that we're asked to calculate and were also asked to calculate what this angle of deflection is. And then after the collision, the Falcon has bounced back and is now traveling at a rate of 5 m per second. Okay, so any time that you have a collision going on, most likely you're going to be using conservation of momentum. So that's what we're going to be executing here. Um, sorry. This should be enough. So we want the mo mentum, um, of the system. So the raven and the Falcon initially to be equal to the momentum of the system. Finally, So we're going to start off by just writing that equation. And when you have a problem in two dimensions or motion in two dimensions, you want to write a conservation of momentum equation for the extraction, a swell as the y direction. So I'm going to use, um this as my coordinate system. So I'm going to say this direction is positive. Why? And this direction is positive. X and I'm just gonna go ahead and write this formula. So I'm going to start with the X direction. So the final moment, um, in the extraction needs to be equal Thio the initial momentum in the X direction now, Initially, the only thing that's moving in the extraction is the rave in. And so we only need thio calculate the mo mentum from the Ravens. So we're going to dio, um r v r i and that's gonna equal to the final Momenta. Um, in the extraction again, The raven is the only thing that is moving in the extraction. So we're going to dio m r VR f but we need to take the component of V R f. It is in the X direction. So specifically, we're going to be looking at this components says V R f but the X component and then this one is V R f but the y component. So if you analyze that triangle there, you're going to see that v r f X over VRs that's opposite over high partners. Sorry. Adjacent over I partners that's gonna be equal to Coast data. So v r F x is we are f coast data. So that's what we're gonna use here because that's the X component of the velocity that we need. Okay. And then we can also create a similar equation for the Y direction. So we're gonna have Okay, pf, why is equal Thio p I? Why? So the final momentum is equal to the initial momentum in the Y direction. Initially, the only thing moving in the Y direction is the Falcon. So we're just gonna right MF v f i Its motion is completely in the y direction. So we just need to dio sometimes v And then for the for the final momentum in the Y direction we need Thio take into account um the Falcon again It's motion is completely in the y direction. So we can just do m times v And then we're going to multiply that by the y component of the mo mentum for the rave in. So that's going to be m times V. But signed data this time for similar reasoning as the extraction So we've got M R V R F signed data. So now that we've got these two equations, let's go ahead and plug in our knowns here and we'll see what we've got to work with. So in the X direction, I'm going to cancel the M R. Because that's on both sides, and I can get rid of that. Um and then I don't know we are f or theta, but I do know VR I, um, is equal to nine. So this becomes a formula with two unknowns. That's equal to 9 m per second. Okay, so I can't really do anything with that. Let's go ahead and move to the Y direction. Um, so we've got 0.6 kg because that's 600 g for the Falcon Mass times. Its initial velocity is 20 m per second. And then over here got the falcon mass times the final speed. Now the final speed. We have to put a negative in front of it because it has rebounded and it's going in the Y direction. After the collisions, it's going to be negative 5 m per second and then adding onto that, we're gonna have 1.5 times e Got a room here. V r F signed data. Okay, So similar situation we've got here. Um, we have to, uh, to unknown is actually the same two unknowns. Um And so since we have two equations, we can go ahead and, um, solve this completely. So what I'm gonna do is I'm just gonna work with the second equation here. So, um, points six times minus 5 m per second. I'm going to bring that over to the other side. Everything else can stay. So when I bring that over, that minus is going to become a positive. And so I get 0.6 times 20 plus 0.6 times five, and that's going to give me 15. And then I'm going to divide by the 1.5 on the left hand side. And so that's going to give me BRF. Signed data is equal to 10. So now we've got two equations, um, relatively simple equations, and we can go ahead and solve for data by dividing the two equations. So I'm going to take, um, do that in a different colors. So I'm going to take VR f Signed data is equal. Thio 10 and I'm going to divide that by V. R F coast data equal to nine. So I get something like that the V R F cancels, and then I'm left with signed data over Coast data, which happens to be tanned data. So tan theta is equal to 10/9, and then we'll just take the tan inverse of that to get data data is equal to tan inverse of 10/9. And when you plug that in your calculator, you're gonna get 48 degrees. So that is the answer to, Let's say, part A yep, part A. That is the direction by the angle by which the Falcon is able to change the Ravens direction. And then we can go ahead and sub that into either one of the equations. So V. R F signed data is equal to 10. So that means V R F is equal to 10 over. Signed Fada is signed 48 degrees, and that will give us a final speed for the rave in of 13.46 meters per second. So that is our final answer for part beat. So this is part A, and here we have party