Vectors and Projectiles Legacy Problem #34 Guided Solution

This is a very difficult angled launch projectile problem, and like any difficult problem, you would do well to employ the habits of an effective problem solver. That means you read the problem carefully, you get a good mental picture about what's going on, you identify the things which are known, the things which are unknown, and then you use your understanding of concepts and mathematical relationships in order to think through a strategy to get from the known information to the unknown information. So I'm going to begin thinking about this as a football that's going through the air, and I would do well just to draw myself that little parabolic trajectory. Starts on the ground, goes up, and then comes back down, lands on the ground, get a parabola going there, and what we know is we know the time that it's in the air. We know total time, t equals 6.2 seconds. We know the angle of launch, theta, 74 degrees, and that theta, 74 degrees, is the same theta that you find in your so-called box flow equations, the Vox equals Vox cosine theta, Voy equals Vox sine theta equations. Now, that's all we explicitly stated here. We know a whole lot more than that, though. We know for any projectile, Ax is zero, Ay is negative 9.8 meters per second squared, and we know an awful lot even more than that. For instance, we know the time it takes a projectile to rise to the peak point is 3.1 seconds. Time to fall is 3.1 seconds. That's where that 6.2 second hang time comes from. And we also know that when it finally gets to the peak of its trajectory, that the vertical velocity, Vy, is zero at that instant. So if I were to just analyze the first half of its trajectory, from ground or punt point to the peak position, then what I would have is three known variables. t is 3.1 seconds, Ay is negative 9.8 meters per second per second, and V final y is zero. So I could calculate lots from that. I could calculate the vertical distance it is when it gets to that point, which isn't of much usefulness here. And I could also calculate the original velocity, that is, when launched. And that would be very useful for me because the first question asked me to determine the speed at which the ball was punted. So if I can just get the original y velocity, I can use a Vox buoy, particularly the buoy equation, to calculate the original velocity. And if I can get the original velocity, I can get Vox, and I can use the equation dx equals Vox times t, t being given and Vox being calculated from the part a answer. So that's my strategy. Now let's go about doing it. I'm going to pick the equation that goes Vfy equals Voy plus Ayt. I'm going to pick that equation. And I'm going to use my Vfy zero, so I'll say zero is equal to Voy plus negative 9.8 times 3.1. And I can put in my numbers there, and I can solve for V original y. And when I do that, I end up getting 30.38 meters per second, or thereabouts. That's the original y velocity. That's the Voy in the Vox buoy equation, the buoy equation that goes Voy equals V original times the sine of 74 degrees. So I can now solve for Vo, and that's part a of the equation. And when I do my solution, I get about 31.6043 as my value for Vo. Now I can take the equation Vox equals Vo times the cosine of 74 and solve for Vox. And once I get that, it's one easy step to get to dx. dx equals that Vox value times the 6.2 seconds total time through the air. And that's my answer.