A shower head has 20 circular openings, each with radius 1.0 mm. The shower head is connected to a pipe with radius 0.80 cm. If the speed of water in the pipe is 3.0 m/s, what is its speed as it exits the shower-head openings?

Video transcript

Hey everyone in this problem, we have a sprinkler head with 41 openings used to Sprinkle water. The openings are 1.5 mm wide. The Sprinkler Head is supplied with water from a 1" diameter hose pipe. If water flows at two m per second in the hose, were asked to calculate the speed of the water jets leaving the sprinkler head. Now we're asked to find speed of water through some particular part of this sprinkler. We're told about the speed of water in a different part of the sprinkler, okay, in the hose and we're told about the diameter of the hose and the diameter or the size of the openings in the sprinkler head. So let's recall our continuity equation. We have a one v. One is equal to a two v. two. And so the product of the cross sectional area, times the speed of water flowing through that part of our system is going to be equal to the cross sectional area times the speed of water flowing through a different part of our system. And so in this case we have to kind of parts that we're looking at, We're looking at the host and we're looking at the sprinkler head. And so we have that the area in the hose, cross sectional area, times the speed of water in the hose is going to be equal to the cross sectional area of the sprinkler head, or the jets leaving the sprinkler head times the speed of water flowing through that sprinkler head. Alright, so what we want to find is the speed of the jets leaving the sprinkler head. And so we're looking for V. S. So we can isolate for V. S. We're gonna have a H over A. S. Times V. H. What is the cross sectional area of the hose? Will the cross sectional area of the hose is going to be a circle? I recall that the area of a circle is pi times are and in this case it's our of the hose squared. And then we have the area of this sprinkler head, are those openings in the sprinkler head? Now each opening is going to be a circle as well. And so the area of an opening is going to be pie R. S square. But we have 41 openings. And so the total area of cross sectional area where we have water flowing through in that sprinkler head is going to be 41 times pi R. S squared. Okay, so 41 openings each with a cross sectional area of pi R. S squared and then we have all of this times B. H. Now we're given information about our hk were given the diameter of the hose in inches, were given information about R. S 1.5 millimeters for the diameter in millimeters. So we want to convert the inches into millimeters so that these units are the same. And then when we divide our h squared by rs squared, those units will cancel out. Okay, so we could convert everything to inches or millimeters or any other unit either of those is fine. We're going to choose to convert two millimeters and so the diameter D H. Is equal to one And this is going to be equal to one Times 25.4 mm/ in. The unit of inch cancels or divides out. And we're left with 25.4 millimeters. Okay, so to go from an inch to a millimeter we multiply by 25.4. And so we have that are diameter of the hoses 25. mm. Now putting this into our equation, we have pie. Okay, in the numerator and denominator, those will divide out. We have r squared. Well r is going to be half of the diameter So we're gonna have 25.4 mm divided by two, All squared over 41 Times. And again we have 1.5 mm for the diameter of the sprinkler. And so we're gonna have to divide by two as well, 1.5 mm divided by two to get the radius squared And then the speed of water flowing through the hose two m/s. All right, so that looks a bit messy but it's not too bad. We're just dividing and multiplying and we are going to get 13. m per second. When we work this out. Okay, the unit of millimeters squared are going to divide out like we already mentioned. and we're left with meters per second. And that is it. That's what we were looking for. That's the speed of the water jets leaving the sprinkler head. And if we go back up to our answer traces, We can see that if we brown this, we are going to have answer a the speed of the water jets leaving the sprinkler head is 14 m/s. Thanks everyone for watching. I hope this video helped see you in the next one.