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(Score for Question 3:points)Aurora hita baseballwith an initialvelocity of 70 feetpersecondat an angle of 30' withthe horizontal The ball hit her bat when the...

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(Score for Question 3:points)Aurora hita baseballwith an initialvelocity of 70 feetpersecondat an angle of 30' withthe horizontal The ball hit her bat when the ball was feet bove the ground_No one interferes with the ball. How long does take the ball to hit the ground? Round your answer to the nearest hundredth of second_ Show all your work How far did the ball traveb horizontally? Use your answer from Part (a) in your calculations Round your answer to the nearest tenth of foot: Show all yo

(Score for Question 3: points) Aurora hita baseballwith an initialvelocity of 70 feetpersecondat an angle of 30' withthe horizontal The ball hit her bat when the ball was feet bove the ground_ No one interferes with the ball. How long does take the ball to hit the ground? Round your answer to the nearest hundredth of second_ Show all your work How far did the ball traveb horizontally? Use your answer from Part (a) in your calculations Round your answer to the nearest tenth of foot: Show all your work Answer:



Answers

The figure shows the path for a baseball that was hit with an initial velocity of 150 feet per second at an angle of $35^{\circ}$ to the horizontal. The ball was hit at a height of 3 feet off the ground. a. Find the parametric equations that describe the position of the ball as a function of time. b. Describe the ball’s position after 1, 2, and 3 seconds. Round to the nearest tenth of a foot. Locate your solutions on the plane curve. c. How long is the ball in flight? (Round to the nearest tenthof a second.) What is the total horizontal
distance that it travels, to the nearest tenth of a foot, before it lands? Is your answer consistent with the figure shown? d. Use the graph to describe something about the path of the baseball that might be of interest to the player who hit the ball. Then verify your observation algebraically.

So in this problem, we are given a set of information. So we know that a a ball is kicked at an angle and it lands 350 feet away about four seconds later. And so we want to know the initial velocity of the ball. So what? So what do we know? Well, we know we know that acts of 04 is equal to is equal to 350 feet. So in other words, the distance travelled after four seconds is 350 feet. And so if we know that we know that except for is 3 50 well, we're gonna use this equation. So these are the parametric equations of E a position of an object on Earth. So So we account for velocity. We account for the angle of the velocity as well as the acceleration Earth. But in this case, we only are given the the total distance traveled. So we're gonna use this x X function for a position. And so, if you know, the distance travelled is 3 50 feet after four seconds. Well, what we have the not and we know the angle. The angle is also given to us. So the angle is is 30 degrees. So the nod co sign 30 times, times 44 seconds. And our initial X position is zero since we're starting at the ground and at a given position. So we have here that we we know everything in this equation except for every not so we can solve for be not and we say Okay, he not is equal to 3 50 3 50 over four co sign 30 over four coastline 30. And so we want to know V not to the nearest whole number. So if we plug this into our calculator, what we get is what we get is 3 50 divided by four co sign of 30. And this gives us 101. So 101 And we won't make sure we include our units. 101 meters per first, our feet for seconds. That's we're dealing with with feet. So 101 it her second. So this is our This is our initial velocity

He's going to use the range equation for party. This is equaling the initial squared times Sign of tooth Ada divided by G This is gonna be equal to the ex velocity times t and to lend me we know that the X component of the velocity is gonna be the initial velocity times co sign If ada and so we can plug in and say that then solving for Fada, we can say that data would be equaling 1/2 arc sine of RG divided by the initial squared and we can solve this would be 1/2 arc sine of the range 7.0 meters multiplied by 9.8 meters per second squared. This would be divided by the initial velocity, 12.0 meters per second quantity squared and so we find that data is equaling approximately 14.2 degrees. Therefore, we know the ball is thrown at an angle equaling 14.2 degrees. This would be your answer for part a now for part B, the other angle that could give the exact same um, the other angle, which exactly which essentially can give the exact same range can be calculated from the complimentary angle so we can say fate of prime would be equaling 2 90 degrees minus data, and this would simply be 75.8 degrees. Now we can say that the time of flight again for part being the time of flight is equaling the range over the ah, the X component of velocity. So v initial times co sign Fada. And so here we can say that we're going to substitute T. This would be equals 27.0 meters, divided by 12.0 meters per second times co sign of 14.2 degrees. This is giving us 0.602 seconds or we can say that T is equaling 7.0 meters, divided by 12.0 meters per second times co sign of 75.8 degrees. So it depends on theta, and this would be equaling 2.3 eight seconds. Um so here we can say that it would not be used. Ah, the 75.8 degree angle is not used because the time of flight for this case is longer than the time of flight for the angle 14.2 degrees. This implies that the defense player will get enough time to get into position to knock down the pass of the ball easily. So if t actually indeed equaled 2.38 seconds, then the past wouldn't be able to be. Wouldn't, um, actually be able to be completed so we can say 75.8 degrees is not used because she is too large. So I would be your answer for parts be. And, um, when this says for part C, how long did this past take? Well, we already calculated it. T would be equaling 0.6 02 seconds. This would be first data equaling, equaling 14.2 degrees and then T is equaling 2.38 seconds. This would be for theta equaling 75.8 degrees. This would be your answer for part C. That is the end of the solution. Thank you for one

In this example will be looking at a projectile problem in which we threw a ball to someone else. Ah, an initial speed of 12 meters per second and this ball will travel a total distance that will call our for the range of seven meters. Now the first thing we're gonna figure out is what the possible angles of these trajectories could be, cause we could throw it in different ways, right? You could imagine we could throw it kind of straight to our friend or we could throw it up high into the air and have, ah, much higher trajectory. So we're gonna figure out what those possible angles are given that we know what we know. And after that, we're going to choose which one of our trajectories we would rather use, um, given the two possible ones. And after that, we're gonna figure out how long it takes for the ball to fly along at your get so to determine what the angle is going to be. We're just going to use the equation for the range of a projectile because it involves the initial velocity, the range and the angles data, which are all things were interested in, and one of which is are unknown, right being data. So the equation for the range of projectile is gonna be our equals B not squared times the sign of two fatal over cheap. So we can just go ahead and solve this for the sign of two fatal right away. So we're told by both sides by G and divide both sides by view, not squared. Well, sign of two feet on the ring inside, and then we can take the sine inverse of both sides like we usually dio so we'll have to Fada equals you sign in verse, but G times are over, you know, squared, and we'll stop here because what we're gonna do is figure out with the possible values for ah, this sign in verse could be. And actually, really, the better way to think about this is to take a step back for equation here and ask ourselves the sign of what angle is going to give us. G times R over V, not square. So we could call this two times data here something else like he may be and ask ourselves, What does he have to be in order to give us g times R over v non square. So this g r over v nonce weird here is going to be equal. Teoh zero point for seven seven It's will have something like the sign of a few years equal Teoh 0.477 So we know for a fact. Ah, this should have multiple solutions, right? Because the sign should be equal to 4.4 point 477 Probably somewhere around here on our unit circle, because we know it's gonna be less than the one for 45 degrees since that one is ah went over square into which is like 0.707 And we know that the sign is going to be identical on the opposite side of our unit circle over here when it's at an angle of 180 minus this angle over here. So that will be. This angle here would be 180 minus data. So there's two solutions to this. The first solution is just going to be given by B sine inverse of 0.477 and then the second solution. Given what we just drew here nearly we should call these fees, since that's what we're working in. The second solution is going to be he sine inverse 0.4777 from get all mixed up here to be 1 80 minus the sine inverse of 0.4777 Okay, so we can just go ahead and solve for both of these, and we know that we put this Ah, fee here is kind of placeholder, but really, we know that he is just two times data, so we can just put that back so we can just put two times They're right here again, so they should be the two solutions. So the first solution, which will be fatal, eyes equal to sine inverse of point for 77 divided by two. He's going to be a 14.2 degrees and then the 2nd 1 just going to be 180 degrees minus sign of 0.477 divided by two should be about 75 20 degrees. Okay, so now the question is, which of these angles would be rather use? And the answer to that question is kind of subjective, but it's that we'd rather used. Yeah, shorter angle here. And the reason is because a shorter thrill angle for a lower throw angle, I should say, will give us a larger component velocity in the X direction, which means that it will travel the distance. We're trying to throw it faster than it would if we did a high lob up, right, So this high lob would take much longer to travel the same distance because the X component of its velocity would be much smaller. So we're going to say that we prefer this one if we want faster passes. So the last thing we want to figure out is that if we take this faster past year, how long it will take our ball to travel seven meters horizontally so we know that we're gonna travel seven meters horizontally, and we're gonna do it with a speed Ah v not acts. Since that's never gonna change right? And that speed is just going to be be not times the co sign of our angle here, you're saying, is the lower angle fatal in And of course, this speed he's just going to be related to the range and the time it takes by the distance over the time. Remember velocities distance over time so we can go ahead and just solve this for Delta t. So multiply both sides by Delta t and divide both sides by the non times. Because China Fada, we'll get the delta T is equal to our divided by v non times that coastline if they know one. So now we have our we have the not and we have the coastline of data one. So we can just go ahead and sell for our time and we will get zero coined six year, a few seconds.

Okay and this problem, we're told that football is kicked and travel 60 ft before hit the ground, but we don't know what she's actually, it could have taken. So we can use some of our old tools from projectile motion questions um and we can write out an equation for why and for us. And we know two things about this problem. One is that at some point the ball hits the ground so y equals zero at the and the ball will travel 60 ft, So that's equal 60. So using our conclusions for X and Y. Um we can solve this system of equations. Well do this using substitution. So we'll substitute for T. 60 over be not so silent data which we got from rearranging this equation. Now we're gonna plug that into this equation and then that's going to spit out. Um This little formula co signed data, signed data equals G. Times 30 over 625. So this is a little bit difficult to solve. So we have to use a trig identity. Um which will give us one half sine of two theta. And this is much easier to solve. So fada is going to be equal to one half because we're driving off this too, the inverse sine of and then here we've got what was on the other side. Um this to gets multiplied up here. So we're gonna have 60 times G over 625. And then when we plug this into a calculator to do the inverse sine, it gives us two different options After we fired by the one house. We get that. Hey, that can either equal she. All right this in bread 35 0.1 degrees. Which would be the red trajectory you see here for If you also equal 54 0.9 degrees. So those are our answers for part A where we just had to find the two potential angles and then in question be they ask us to use these angles to figure out how long the ball could have been in the air. So we can plug these angles back into um our original equation. Um So we can get two equals 60 over um The zero, which is 25 um times recent data. And then if we plug in the two angles will get two answers again and I think either equal 2.9 three three seconds, or it could also equal 4.17 and that's it.


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