Vectors and Projectiles Legacy Problem #29 Guided Solution

This problem falls into the category of a so-called angle launch projectile problem in which the projectile originally starts its motion at an angle to the horizontal. As such, it has both a horizontal velocity and a vertical velocity. We will need to calculate the so-called vertical and horizontal velocities, the VOx and the Voy, using the Vox-Voy equations. The Vox-Voy equations are discussed on the overview page of this problem set. You'll see the link right below at the bottom of this page, and we can calculate Vox using those equations by going 34.9 meters per second multiplying by the cosine of 35 degrees. We can do the same thing with Voy, 34.9 times the sine of 35 degrees. I'd write those two numbers down to several significant digits, and you're going to need to use those numbers in subsequent calculations. Now one of the things I need to do here is determine the total time of flight. Now what we know about our projectile is that it begins its motion and it heads up towards a peak position and falls back down, and that the time it takes to get up to its peak position is equal to time for it to fall from its peak position. We also know that at that very peak position that there's no more y velocity, that By has turned to zero meters per second. I'm going to use that information in order to calculate the time it takes to go to the peak, and then the total time it takes for its flight. So the time it takes to go to its peak can be calculated using the kinematic equation that goes something like this. Vfy equal Voy plus Ay times t. The Vfy, that is the y velocity at the peak, would be zero meters per second, and the Voy would be what you calculated when you use your Vox-Voy equation, which was 20.00178 meters per second. So I take that 20.00178 meters per second for Voy, and I say zero is equal to that number plus negative 9.8 times t, and I solve for t. When I do that calculation, I get a little bit short of a little bit more than 2.04 seconds. I then double the number because the time it takes to go to that peak position can be doubled to get the total time of flight. And when I'm done, I get a number of about 4.0853 seconds. That's the total time of flight, and I write it down to all those digits because you're going to need to use that time in the next calculation. In part B, you're asked to calculate the horizontal displacement. Now, unfortunately, we only have one equation of usefulness to us when it comes to horizontal motion, and that's the equation that dx equal Vox times t. So you know you have to use that equation. Your Vox, you can calculate from your Vox flow equation. It comes out to be 28.5884. And the t is what you just calculated in part A of the problem. It comes out to be 4.0853 seconds. So go tx equal Vox times t, and you get your answer there. It comes out to be about 117 meters. Now we have to calculate part C, which is determine the peak height. The way you get the peak height is you pick an equation that has a dy in it. The one I like to use for this is simply the equation that goes dy equal, and the numerator, Voy plus Vfy, and then in the denominator, 2. So take the sum of the original and the final y velocities and divide by 2, and then multiply by the time it takes to get to the peak position because we're trying to find the peak height. So what's the final velocity at the peak? Well, it's just simply zero. That's the final y velocity. What's the original y velocity? Well, we calculated that with our Vox flow equations, and I hope that you wrote it down. It was 20.00178. And what's the time it takes to go up to the peak? I'm hoping you wrote that one down too. It's simply half the answer to question A. It's 2.0426. So here's how I'm going to calculate that dy, the peak height. I'm going to go dy equal, and then I'm going to sum 0 and 20.00178. That's a pretty easy sum. Then I'm going to divide that number by 2, and then I'm going to multiply it by the time it takes to go to the peak, which was 2.0426. When I'm done, I get an answer of around 20.4, and that's the peak height.