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Total final velocity for projectile

Calculating the total final velocity for a projectile landing at a different altitude . Created by Sal Khan.

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  • blobby green style avatar for user tgsmikema
    isn't the final vertical velocity is -26.03 instead of -29.03??
    (13 votes)
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  • male robot hal style avatar for user Conor McKenzie
    How would you do this using calculus?
    (12 votes)
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  • blobby green style avatar for user Damian Radinoiu
    Can someone explain why the horizontal velocity haven't changed due to acceleration ? Why is acceleration 0 on the horizontal line ?
    (4 votes)
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    • mr pants teal style avatar for user Arshia
      Acceleration is zero on the horizontal because a few videos ago,we actually assume that drag is negligible. So, if drag is neglible and we ignore friction as well, there are no forces acting upon it. If there are no forces acting upon something then, the acceleration is zero.

      This can be seen by f=ma (Newton's Second Law Of Motion)
      So, if F= O, then it doesn't matter what mass is.
      0= ma

      The projectile has mass, but Acceleration will be zero because anything multiplied by zero is zero
      m= O * accn
      So, there is no acceleration, because there are no forces
      (9 votes)
  • male robot hal style avatar for user Alisher  Bakhramov
    There is said:"we figured out... in the last video..." in this video. My question is: In what last video. There were another conditions in previous video "Launching and landing on different elevations".
    (6 votes)
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  • blobby green style avatar for user Mustafa Majormuss Ibrahim
    Why does Sal use the initial component of the velocity of 29.54m/s when it seems it should 0m/s from attaining its highest point? I thought the initial velocity would be from the apex of its trajectory since that is the immediate initial velocity before the projectile lands..
    (5 votes)
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    • male robot hal style avatar for user istvan.gulyas
      We know the time that passed since the launch of the projectile, not the time that passed since it's highest point. So we're using the total change of time, from the very beginning, that's why we also use the launch velocity. You could calculate it from the apex, but then you have to first figure out the change of time from the apex to landing, which will be less than 5.67 s
      (6 votes)
  • duskpin seed style avatar for user Fluff-ball
    Two questions:
    1. We will always take the final velocity vertically when calculating the total velocity?
    2. Why did Sal not take the -26.03 value instead of +26.03??
    (7 votes)
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    • blobby green style avatar for user Wai Yan
      1) Yes, even if it 0m/s. Although, if it were, we wouldn't really need to take it anyways.

      2) 1-dimensional planes have only 2 directions (left, right, etc.) which are described using + and - symbols.

      For 2 dimensions, it becomes complicated because there are multiple directions. Using + and - symbols would not work. So, directions are described using angles from the horizontal line.

      But, regardless of this idea, or maybe because of it, Pythagoras' Theorem is used. And from the start, this theorem is only suited to find the magnitude of the vector. It is not suited to include directions in the calculation.

      (Adding a negative sign presumably gives a false answer. Because in real life, even if we point the horizontal line in the opposite direction, the hypotenuse's length would be the same)

      And finally, when the magnitude is obtained, it can be used alongside the angle/direction to describe the vector.
      (2 votes)
  • blobby green style avatar for user Rupal R. Shah
    At Sal calculates -26.026. When I subtract 9.8 from 29.54 and multiply that by 5.67, I get 111.93. Why do I not get -26.026? Is my math incorrect?
    (2 votes)
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  • male robot hal style avatar for user Siddharth
    For calculating the final vertical velocity, is it possible to use the formula: displacement=((initial velocity + final velocity)/2)*change in time? After substituting all the known values in, I get the result −26.017 which is only slightly different from Sal's result. Could this be because of small rounding errors or is this not the right equation to use?
    (2 votes)
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  • starky tree style avatar for user Mattyb
    How would you do this if you were accounting for air resistance ?
    (1 vote)
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  • blobby green style avatar for user olivia144
    What is the horizontal and why is it the angle below the horizontal? The object lands higher than where it was launched, so I'm confused about why the direction of the velocity would be "below" or negative.
    (1 vote)
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    • old spice man green style avatar for user Rafi Hagopian
      We are talking about the velocity here, not the displacement. The object first went up and, when it hit the top platform, it's on its way down again. Therefore the direction of its movement over time (definition of velocity) is downward, with that angle. Hope this helps.
      (3 votes)

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.