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Anderson Video - Rocket Loses Bolt on Takeoff

Professor Anderson
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 >> I mean we kind of know what it is, right? It’s the rocket at four seconds, a bolt falls off And it falls at seven point one seconds and hits the ground. At what cut speed was the rocket traveling at that at that time? >> Okay, so let's let's see if we can just draw a picture to start and then we'll worry about what exactly it's gonna look like in a second. And this thing is going up and now at t equals four seconds, there is a bolt that falls off and comes back down to Earth like that. Is that right? And what we know is that that happened at t equals four seconds and the total time— so this is t for the rocket and the time for the bolt to fall is seven point one seconds. All right? >> Yes. >> Okay and what are we trying to find? >> Acceleration of the rocket. >> Find the acceleration of the rocket. a sub y for the rocket Okay, is that what we know? Is that right? Okay, let's try this one. Are we still rolling? Hello class, Professor Anderson here. Let's take a look at a problem that some of you are having some trouble with. This is the rocket problem. The idea is that we have a rocket that takes off and four seconds after takeoff, there is a bolt that falls off and the bolt follows this sort of trajectory it rises a little bit and then it falls back down to earth and all we know is that at t equals four seconds the bolt fell off the rocket and seven point one seconds is how long the bolt was in the air. Is that right? Is that the problem that you guys are facing? >> Yes >> Okay, so this is the total time that it's in the air, not at the same clock time seven point one seconds it hits the ground. >> Correct. >> Okay. So what we need to figure out is what the acceleration is of this rocket. This rocket is accelerating upwards at some constant acceleration and we need to figure it out. This looks challenging, right? We don’t know a lot of information here. What sort of information maybe would help us? What do you think? >> The total distance traveled by a bolt. >> Okay if we knew something about the heights here that would be pretty good right so why don't we label a couple heights. This is the initial position of everything. That we That we can call zero. Let's call this height y1 which is how high is the rocket when the bolt comes off. Let's call this height up here y2: how high does the bolt actually get? And let's see if we can make some sense of this. So first off, We have a nice equation that looks like this. yf equals yi plus vyi times t plus 1/2 a sub y t squared. Now, if we're going to apply this equation to the rocket going up, what can we say? Well, after four seconds it has reached a height y1. It started at a height of zero. It started from rest. Rockets typically start from rest on the launch pad. So that's zero. And now we have 1/2 a sub y. That time t sub r is 4 seconds. And so now we have one equation but we still have two unknowns. We don't know y sub one and we don't know a sub y but let's just keep that equation there for a second. What about the bolt falling back to Earth? Can we figure out the motion of that? We could probably use the same equation and look at the motion of the bolt. So if we use that same equation, this was the rocket going up. Let's look at the bolt. Let's use the exact same equation that we have right here. If we do that, what is y final for the bolt? >> Zero. >> Zero, right? Ends up on the ground. What is y initial for the bolt? Where does it start its projectile motion? Well, if we start from right there then it starts at y1. vy initial times t, what is that? Well, however fast the rocket is moving at this point, that's how fast the bolt is moving at that point. So that is v sub r times t sub b for the time that the bolts in the air. And then we have one half times a y, But a y for the bolt is just negative g. Once it leaves the rocket, it's in freefall. And then we have t sub b squared. Okay so let's rewrite this equation a little bit. We've got this quantity which is a negative, let's move that over to the other side. We've got one half g t sub b squared. And that's going to equal y sub one plus v sub r t sub b and let's just identify how many unknowns we have in this equation. We've got g, which we know. We've got t sub b, which they gave us. We have y1 which we don't know and we have v sub r which we don't know. But let's just put a box around this equation and let's identify what are the unknowns at this point. y1, we don't know that. a y, we don't know that. v sub r, we don't know that. Everything else we know. But that's not going to do, it is it? Because we have three unknowns and we only have two equations and so we need one more equation to put all this together. Is there anything else that we can say about some of these quantities? Particularly this v sub r right here. Is there something we can say about v sub r? How fast is the rocket moving when it gets up to this height? Yeah. >> Is it however long it's been accelerating for? So acceleration times time? >> Acceleration times time sounds very good. Let's see, we have a kinematic equation that looks like this: vf equals vi plus at. We know that the a is the acceleration of the rocket and we know that the time is the time that it took the rocket to get up there. And so look what happens, we have v r equals zero is our initial velocity so we can just cross that one out and we end up with a y t sub r. And let's put a box And let's put a box around that. That is the speed of the rocket right there. And so now look what happens? We've got one two three unknowns. We have one two three equations. So now it is just a matter of solving these together to get all these quantities that we want and in particular we want a y. I'm not going to bother you with all the mathematical details at this point this is the underlying physics and at that point it's just doing some math to solve for those quantities. So hopefully that's clear. If not, come see me in office hours. Cheers. Cheers.