You are watering your lawn with a hose when you put your finge...

Question

You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are holding the hose horizontally, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening did you block?

Accepted Answer

Welcome back. Everyone in this problem. While playing with a water gun, you partially cover the nozzle to increase the range of the water stream. Initially, the water stream reaches a certain distance, but by partially covering the nozzle, the range increases by a factor of five. If you're holding the water gun horizontally, what fraction of the nozzle opening did you block? A says it's five minus the square root of five all divided by five B five plus the square root of five all divided by five C, the square root of five divided by five and D says it's 1/5. No, let's make sure we understand what we're trying to figure out here. So we want to find a fraction of another opening that we blocked for the range to increase by a factor of five. If we try to visualize what's going on here, then we're saying initially, if, if we picture this as the nozzle of the water gun, when we shoot from the water gun or from this nozzle where none of it is obstructed, OK, then the water stream covers a range of R, OK. Let's call that R a distance of R but no for the, for the same nozzle. Ok. We're saying that when we cover a part of it, we don't know how much we cover. Ok. Let's say that this is the fraction that we have covered. And now the amount of water that's coming out is this is, is represented by that shaded blue area, then the range is now going to increase by a factor of five. In other words, the range is now five R, the distance is not five R. And we want to figure out what fraction of the nozzle opening we blocked. In other words, we're trying to figure out this shaded part of our diagram. So let's call the area for our first nozzle A one that is coming from and for a second nozzle, let's call that area A two. Then essentially what we're trying to find here is going to be one minus, OK, one minus A two divided by A one because we're finding what fraction of the original area is able to spray water. And we're going to subtract that from the hole to figure out then the part that we had blocked. So how are we going to figure out what fraction was blocked? Well, recall that we know in the continuity equation that area multiplied by the speed in our first scenario is going to be equal to the area in the second scenario multiplied by the speed in the second scenario. Now we know here that our water stream it takes on the path of a projectile. OK. And recall that for the range of our projectile R, it's equal to V squared multiplied by a sin to theta divided by G. Now, we're not going to get into all of this to figure out the actual range. But from this equation, we can tell an important relationship here, we can see that the range is directly proportional to the square of the speed. So what that tells us then for our horizontal protection projection, since the range increased by a factor of five, that means the square of the speed also increased by a factor of five. In other words, this tells us that Z two squared then must be equal to five multiplied by V one squared. And if that's the case, that means V two is going to be equal to the square root of five multiplied by V one squared, which is going to be root five multiplied by V one. Now, why is this important? Well, using this relationship, notice that from our continuity equation, we had an expression for a two and V two on one side. But no, we can rewrite V two in terms of V one and thus find our ratio of a two divided by a one and subtract it from one to figure out a fraction of the nozzle that is blocked. OK. Now, that means then substituting this into our continuity equation. Then a one, V one is going to be equal to a two multiplied by root five, multiplied by V one. Ok. We can cancel V one from both sides and divide both sides by a one and, uh, the route of five. But you know what, let me write that thought actually. So we can divide this side by a two multiplied by root of five here. But no, we can divide, you know what? Let's just do the math. Let's just do the math. Let's do it one by one. So we can divide through by a two. OK? So now we'll get a one divided by a two being equal to the square root of five. And now to find a two divided by a one, we can reciprocate to get it to be equal to one divided by the square root of five. If we rationalize the reciprocal of the square root of five, that means we can multiply by root five divided by route five, which now will leave us with route five divided by five. Now that we have a value for the ratio of a two to a one, we can find a fraction that was blocked. So therefore, the fraction blocked. OK is going to be equal to one minus root five divided by five, which if we write that as one fraction, it's going to be five minus root five divided by five. So we could use this expression five minus root five all divided by five to figure out a fraction of the nozzle opening that is blocked. If we go back to our answer choices that tells us, then that a is the correct answer. Thanks for watching everyone. I hope this video helped.

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You are watering your lawn with a hose when you put your finger over the hose opening to increase the distance the water reaches. If you are holding the hose horizontally, and the distance the water reaches increases by a factor of 4, what fraction of the hose opening did you block?

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