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Calculating the total final velocity for a projectile landing at a different altitude (mistake near end: I write 29.03 when it should be 26.03 m/s and the final total magnitude should be 26.55 m/s 78.7 degrees below horizontal. Created by Sal Khan.
Video transcript
In the last video, I told you that we would figure out the final velocity of when this thing lands. So let's do that. I forgot to do it in the last video. So let's figure out the final velocity-- the vertical and the horizontal components of that final velocity. And then we can reconstruct the total final velocity. So the horizontal component is easy, because we already know that the horizontal component of its velocity is this value right over here, which we-- this 30 cosine of 80 degrees. And that's not going to change at any point in time. So this is going to be the horizontal component of the projectile's velocity when it lands. But what we need to do is figure out the vertical component of its velocity. Well, one thing we did figure out in the last video, we figured out what the time in the air is going to be. And we know a way of figuring out our final velocity from an initial velocity given our time in the air. We know that a change in velocity -- and we're only dealing in the vertical now-- we're only dealing with the vertical, because the horizontal velocity is not going to change. We've assumed that air resistance is negligible. So we're only dealing with the vertical component right over here. We know that the change in velocity-- or, we could say the horizontal-- the vertical component of the change in velocity, is equal to the vertical component of the acceleration times time. Now, we know what the change in time is, we know it is-- I'll just write down times our time. And what is our change in velocity? Well, our change in velocity is our final vertical velocity minus our initial vertical velocity. And we know what our initial vertical velocity is, we solved for it. Our initial vertical velocity, we figured out, was 29.54 meters per second. That's 30 sine of 80 degrees, 29.54 meters per second. So this is going to be minus 29.54 meters per second, is equal to-- our acceleration in the vertical direction is negative, because it's accelerating us downwards, negative 9.8 meters per second squared. And our time in the air is 5.67 seconds. Times 5.67 seconds. And so we can solve for the vertical component of our final velocity. So once again, this is the vertical component. This isn't the total one. So, the vertical component. Let me-- well I wrote vertical up here. So there's the vertical component. So let's solve for this. So if you add 29.54 to both sides, you get the vertical component of your final velocity. Well, this is a vertical component, I didn't mark it up here properly-- is equal to 29.54 meters per second plus 9.8 plus -- or I should say minus-- meters per second. Minus 9.8 meters per second squared, times 5.67 seconds. The seconds cancel out with one of these seconds. So everything is meters per second. And so, get the calculator out again, we have 29.54 minus 9.8 times 5.67. So we get our change-- our final velocity is negative 26.03. So this is negative 26.03 meters per second. And you might say wait, wait Sal , what is this negative 26.03 meters per second mean? Remember, when we're dealing in the vertical dimension, positive means up, negative mean down. So it means we're going 26.03 meters per second downwards. Downwards, right when we land. So what is our total velocity when we fall back to that landing? So the vertical component of our velocity is negative 29.06 times .03 in the downward direction. And the horizontal component of our velocity, we know, hadn't changed the entire time. That, we figured out, was 30 cosine of 80 degrees. So that over here, is 30 cosine of 80 degrees. I'll get the calculator out to calculate it. 30 cosine of 80 degrees, which is equal to 5.21. So this is 5.21 meters per second. These are both in meters per second. So what is the total velocity? Well, I can do the head to tails. So I can shift this guy over so that its tail is at the head of the blue vector. So it would look like that. The length of this-- the magnitude of our vertical component, is 29.03. And then we could just use the Pythagorean theorem to figure out the magnitude of the total velocity upon impact. So the length of that-- we could just use the Pythagorean theorem. So the magnitude of our total velocity, that's this length right over here. The magnitude of our total velocity, our total final velocity I guess we can say, is going to be equal to-- well that's-- let me write it this way. The magnitude of our total velocity is going to be equal to square root-- this is just straight from the Pythagorean theorem-- of 5.21 squared plus 29.03 squared. And we get it as being the second-- the square root of 5.21 squared plus 29.03 squared gives us 29.49 meters per second. This is equal to 29.49 meters per second. That is the magnitude of our final velocity, but we also need to figure out its direction. And so we need to figure out this angle. And now we're talking about an angle below the horizontal. Or, if you wanted to view it in kind of pure terms, it would be a negative angle-- or we could say an angle below the horizontal. So what is this angle right over here? So if we view it as a positive angle just in the traditional trigonometric way, we could say that the-- we could use any of the trig functions, we could even use tangent. Let's use tangent. We could say that the tangent of the angle, is equal to the opposite over the adjacent-- is equal to 29.03 over 5.21. Or that theta is equal to the inverse tangent, or the arctangent of 29.03 over 5.21. And that gives us-- we take the inverse tangent of 29.03 divided by 5.21, and we get 79.8 degrees. But it's going to be 79.8 degrees south, or, I guess, below the horizontal. Or you could view this as an angle of negative 79.8 degrees above the horizontal, either one of those work. What's neat about this, is we figured out our final velocity vector. The entire vector, we know what that entire vector is. It is 29.49 meters per second at 79.8 degrees below the horizontal.