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.