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2-dimensional momentum problem (part 2)

We finish the 2-dimensional momentum problem. Created by Sal Khan.

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  • leaf green style avatar for user Tom Ford
    At - Is it necessarily true that having a smaller mass will mean it (ball B) is deflected at a larger angle?

    Does having a smaller mass result in a larger degree of deflection? To me I think it is more complicated than that.
    (31 votes)
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    • leaf green style avatar for user Vu
      Without doing any rigorous proof, I'll try to explain it.

      1/ "An object at rest tends to stay at rest."
      Initially, both balls have no y-component value, thus they are at rest in the y direction, thus they will tend to resist moving in the y direction.

      2/ We also know F = ma -> a = F/m. Essentially, this tells us that the mass is inversely proportional to the acceleration (or change in movement, so to imagine). This translates into "The more mass something has, the higher its tendency to maintain its motional state."

      From the above 2 premises, we can conclude that both ball A and ball B will resist their movement in the y direction (the deflection angle), but ball A will be more successful due to its higher mass.
      (78 votes)
  • mr pants teal style avatar for user Udbhav Dalal
    Why is momentum always conserved?
    (8 votes)
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  • leaf blue style avatar for user Stuffe
    How is energy conserved in this system? I get that you have momentum conserved because the Y components cancel out.
    But it seems like if you took the kinetic energy of the balls separately and added them together you would have a greater amount of energy than there was before?
    (6 votes)
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    • piceratops ultimate style avatar for user Cody Max Eisenhardt
      Assuming that there is no difference in potential energy, the change in energy would be completely accounted for by kinetic energy.
      The kinetic energy of ball A initially is (1/2)mv^2 = (1/2)(10)(3^2) = 45J
      The kinetic energy of ball B initially is (1/2)mv^2 = (1/2)(5)(0^2) = 0J.
      So, initial kinetic energy = 45 + 0 = 45J.

      Final kinetic energy for ball A = (1/2)(10)(2^2) = 20J.
      Final kinetic energy for ball B = (1/2)(5)(3.2^2) = 25.6J.
      Final kinetic energy = 20 + 25.6 = 45.6J (but wait!)

      Now it may seem that kinetic energy has increased, but Sal has been rounding his values and this extra 0.6J can be wholly attributed to rounding errors. It is completely safe to say that kinetic energy is conserved. If you re-do the problem with a good calculator and take values at 4 (or so) decimal places, you'll get a value of 45J for the final kinetic energy
      (12 votes)
  • blobby green style avatar for user koenige14
    at how do you know momentum is conserved? does it tell you this in the problem??
    (9 votes)
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    • blobby green style avatar for user Raunaq Arora
      What everone will tell you is that momentum is not conserved only when an external force acts on the system.
      It's not a matter of what forces are acting, it's just a matter of including all object exerting forces in the system. For example, (2 pendulums, one held at 90o to the left and one held at 90o to the right. They are let go and collide at the bottom (zero degree). is momentum conserved? the answer is no because gravity is acting on both pendulums.) the two pendulums have gravity acting on them. That's exerted by the Earth itself. Either you need to include the Earth in the system and account for the transfer of momentum from the pendulum to the Earth or you have to consider the Earth as an external force. The former has no external forces and has a constant momentum, the later does not since gravity is an external force.
      Think about a canary in a cage, then let the canary out of the cage but in your room. If you treat the cage as the system, the canary is not preserved. But if you expand your system to include more objects (your room), the canary is preserved.
      To sum it up, if momentum is not conserved, your system is not big enough.
      (4 votes)
  • leaf green style avatar for user Azmath Rahiman
    Initial momentum of A + Initial momentum of B = 30. Final moentum of A + Final momentum of B = 20 + 16 = 36. Now where does this 6 additional momentum come from and how is it conserved?
    (5 votes)
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    • blobby green style avatar for user a.bernicchia
      It seems you simply added the Pa and Pb vectors module and this is not the correct way to add 2 vectors, especially if they have different directions.
      Initial P vector is: Pax+Pay+Pbx+Pby=10*3x+0y+0x+0y=30x+0y
      Final P vector is: Pax+Pay+Pbx+Pby=10*sqrt(3)x+10*1y+2,54*5x-2*5y=30x+0y
      (6 votes)
  • aqualine ultimate style avatar for user Tanis Kint
    Okay, so this is just a weird thought - if momentum is conserved, does that mean that the universe, taken as a system, has a momentum of zero (or some other such constant)?
    (5 votes)
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    • leaf green style avatar for user Pranjal Gupta
      Yes it does.But we dont know whats out of the universe do we?So if there was something outside the universe , the momentum would not be conserved(not equal to zero).(since the universe and the object outside would have a gravitational force of attraction.)
      (3 votes)
  • male robot hal style avatar for user Conor McKenzie
    Throughout the question, we assume that ball A is deflected at 30 degrees. Would ball A actually deflect at 30 degrees? If not, how could we find the actual angle it would deflect at?
    (3 votes)
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  • blobby green style avatar for user nnnamuujin
    couldn't we just use tan^-1? on @6.44?

    (1 vote)
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  • starky sapling style avatar for user saintflemme
    I'm taking a physics class right now and we had an experiment just like that. The output angle between the two balls is ALWAYS 90 DEGREES, something must be wrong with thoses numbers Sal is inputing because the angle between the two balls at output can't be 68 degrees.
    (1 vote)
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  • aqualine seed style avatar for user Anthoan Oroz
    I cant completely understand what is momentum? is it another form of potentional energy converting to kinetic and vice versa?

    Also, this is a stupid qquestion but im gonna ask it anyway, if a skier accelerates from a hill downwards for a while, has he a momentum or a force? because
    F = ma but momentum is p=mv

    Sorry for bad english.
    (1 vote)
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    • male robot hal style avatar for user Andrew M
      No, momentum is not energy, it is its own thing. It is mass*velocity. You can think of it as how hard it is to stop an object that is in motion. A big heavy object is hard to stop once it is moving. A light object is easier. The light object has less momentum.

      Also, It is harder to stop the big object if it is moving faster. than when it is moving slow. It has more momentum.

      Momentum is useful because it is conserved.

      Newton's law F = ma was originally formulated with momentum: F = rate of change in momentum
      (4 votes)

Video transcript

Welcome back. When I left off I was rushing at the end of this problem because I tend to rush at the end of problems when I am getting close to the YouTube 10 minute limit. But I just wanted to review the end of it because I feel like I rushed it. And then, actually continue with it and actually solve for the angle and then, introduce a little bit of-- a little more trigonometry. So just to review what we did, we said momentum is conserved and in two dimensions that means momentum is conserved in each of the dimensions. So we figured out what the initial momentum of the entire system was and we said, well, in the x direction, the initial momentum-- and all the momentum was coming from the ball A right. Because ball B wasn't moving, so its velocity was 0. So its momentum was 0. So ball A in the x direction and it was only moving in the x direction. So it's momentum in the x direction was 3 meters per second times 10 kilogram meters per second. And we got 30 kilogram meters per second. And then there was no momentum in the y direction. And then we knew that well after they hit each other, ball A kind of ricochets off at a 30 degree angle at 2 meters per second. We used that information to figure out the x and y components of A's velocity. So A's velocity in the y direction was 1 meter per second and A's velocity in the x direction was square root of 3. And we used that information to figure out A's momentum in each direction. We said well, the momentum in the y direction must be 1 meter per second times A's mass, which is 10 kilogram meters per second. Which I wrote-- what I wrote here. And then we figured out A's momentum in the B direction and we said well, that's just going to be square root of 3 times 10. And that's 10 square root of 3. And then we used that information to solve for B's momentum. Because we said well, B's momentum plus A's momentum in the x direction has to add up to 30. This was the x direction before. And we knew that B's momentum plus A's momentum in the y direction had to add up to 0, right? And so, since y's momentum going upwards was 10 kilogram meters per second, we knew that B's momentum going downwards would also have to be 10 kilogram meters per second. Or you could even say it's negative 10. And we figure that out based on the fact that B had half the mass. That its velocity going down was 2 meters per second. And similarly, we knew that A's momentum in the x direction, which was 10 square root of 3 kilogram meters per second, plus B's momentum in the x direction is equal to 30. And then we just subtracted out and we got B's momentum in the x direction. And then we divided by B's mass to get its velocity. Which we got as 2.54. So that's where I left off and we were rushing. And already, this gives you a sense of what B is doing. Although it's broken up into the x and y direction. Now if we wanted to simplify this, if we wanted to kind of write B's new velocity the same way that the problem gave us A's velocity, right? They told us A's velocity was 2 meters per second at an angle of 30 degrees. We now have to use this information to figure out B's velocity and the angle of it. And how do we do that? Well this is just straight up trigonometry at this point, or really just straight up geometry. Let me clear all of this. And let's remember these two numbers, 2.54 and minus 2. So B, we learned that in the x direction its velocity-- this is all for B-- is equal to 2.54 meters per second and then y direction, it was moving down. We could write this as minus 2. But I'll just write this as 2 meters per second downwards. Right? Same thing. Minus 2 up is the same thing as 2 meters per second down. So the resulting vector's going to look something like this. When you add two vectors you just put them-- put the one's end at the beginning of the other-- put them front to end, like we did here. And then you add them together and this is the resulting vector. And I think you're used to that at this point. And now we have to figure out this angle and this side. Well this side is easy because this is a right angle, so we use Pythagorean theorem. So this is going to be the square root of 2.54 squared plus 2 squared. And what's 2.54 squared? 2.54 times-- whoops. 2.54 times 2.54 is equal to 6.45. So that's the square root of 6.45 plus 4, which equals the square root of 10.45. And take the square root of that. So that's 3.2, roughly. So the resulting velocity in this direction, whatever angle this is, is 3.2 meters per second. And I just used Pythagorean theorem. So now all we have to do is figure out the angle. We could use really any of the trig ratios because we know all of the sides. So I don't know, let's use one that you feel comfortable with. Well let's use sine. So sine of theta is equal to what? SOH CAH TOA. Sine is opposite over hypotenuse. So the opposite side is the y direction, so that's 2, over the hypotenuse, 3.2. So 2 divided by 2 divided by 3.2 is equal to 0.625, which equals 0.625. So sine of theta equals 0.625. And maybe you're not familiar with arcsine yet because I don't think I actually have covered yet in the trig modules, although I will eventually. So we know it's just the inverse function of sine. So sine of theta is equal to 0.625. Then we know that theta is equal to the arcsine of 0.625. This is essentially saying, when you say arcsine, this says, tell me the angle whose sine is this number? That's what arcsine is. And we can take out Google because it actually happens that Google has a-- let's see. Google actually-- it's an automatic calculator. So you could type in arcsine on Google of 0.625. Although I think the answer they give you will be in radians. So I'll take that answer that will be in radians and I want to convert to degrees, so I multiply it times 180 over pi. That's just how I convert from radians to degrees. And let's see what I get. So Google, you see, Google says 38.68 degrees. They multiplied the whole thing times 180 and then divided by pi, but that should be the same thing. So roughly 38.7 degrees is theta. Hope you understand that. You could pause it here if you don't, but let me just write that down. So it's 38 degrees. So theta is equal to 38.7 degrees. So then we're done. We figured out that ball B gets hit. This is ball B and it got hit by ball A. Ball A went off in that direction at a 30 degree angle, at a 30 degree angle at 2 meters per second. And now ball B goes at 38.-- or we could say roughly 39 degrees below the horizontal at a velocity of 3.2 meters per second. And does this intuitively make sense to you? Well if you remember the problem from before-- and I know I erased everything. Ball A had a mass of 10 kilograms while ball B had a mass of 5 kilograms. So it makes sense. So let's think about just the y direction. Ball A, we figured out, the y component of its velocity was 1 meter per second. And ball B's y component is 2 meters per second downwards. And does that makes sense? Well sure. Because their momentums have to add up to 0. There was no y component of the momentum before they hit each other. And in order for B to have the same momentum going downwards in the y direction as A going upwards, its velocity has to be essentially double, because its mass is half. And a similar logic, although the cosine-- it doesn't work out exactly like that. But a similar logic would mean that its overall velocity is going to be faster than the- than A's velocity. And so what was I just-- oh yeah. My phone was ringing and I got caught up. My brain starts to malfunction. But anyway, as I was saying, so just intuitively it makes sense. B has a smaller mass than A, so it makes sense that-- one, B will be going faster and that it gets deflected a little bit more as well. The reason why it seems like it gets deflected more is because its y component is more. But anyway, that last piece is just to kind of hopefully give you a sense of what's happening and I will see you in the next video.