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Thread: Velocity and kinetic energy

  1. #1 Velocity and kinetic energy 
    Forum Sophomore DarcgreY's Avatar
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    Can anyone explain the relationship between velocity and kinetic energy?

    From what I understand, when you double the velocity of an object the kinetic energy is increased by a factor of four. This seems counter-intuitive to me.


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    Forum Professor river_rat's Avatar
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    Let v be the velocity of a particle measured in some inertial reference frame and for ease of notation lets suppose that one of the coordinate axis are parallel with the velocity vector (i.e. so v = vi for example) . Then the kinetic energy of the particle is KE = mc<sup>2</sup> - m<sub>0</sub>c<sup>2</sup> which for small velocities gives the equation KE ~ 1/2 mv<sup>2</sup>. So if you double the velocity of the particle then KE ~ 1/2 m(2v)<sup>2</sup> = 4*(1/2 mv<sup>2</sup>)

    In other words doubling the velocity increases the kinetic energy by a factor of 4 if the velocities are sufficiently small.


    As is often the case with technical subjects we are presented with an unfortunate choice: an explanation that is accurate but incomprehensible, or comprehensible but wrong.
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    Forum Sophomore DarcgreY's Avatar
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    I wish my math skills were better.

    Could you please expalin what the m0c^2 represents in the equation KE = mc^2 - m0c^2.
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    Forum Isotope Zelos's Avatar
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    einsteins famous equation wich everybody know E=M<sub>0</sub>C²
    C is the velocity of light, about 3*10<sup>8</sup>
    and M<sub>0</sub> is the mass of the object at 0 velocity to a referensframe. and M is the mass of the object when its moving at velocioty v
    M=M<sub>0</sub>/sqr(1-v²/c²)
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  6. #5  
    Forum Sophomore DarcgreY's Avatar
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    Got it, thanks Zelos.
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  7. #6  
    Forum Isotope Zelos's Avatar
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    no problem
    You can rewrite the equation like
    Ek=M<sub>0</sub>C²(1/sqr(1-v²/c²)-1)
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  8. #7 Re: Velocity and kinetic energy 
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    Quote Originally Posted by DarcgreY
    Can anyone explain the relationship between velocity and kinetic energy?

    From what I understand, when you double the velocity of an object the kinetic energy is increased by a factor of four. This seems counter-intuitive to me.
    Why does it seem counter intuitive?

    I personally used to think that too but have come to accept it. Yet it has never seemed intuitive .... then again if it was linear it wouldn't be intuitive either. Hmmmmm ... more questions than answers for me here.

    Pete
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  9. #8  
    Forum Sophomore DarcgreY's Avatar
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    Common sence isn't going to tell you that if you double the velocity of an object the kinetic energy is going to be four times. Early scientists like Galileo were suprised when they discovered this relationship through experiments.
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  10. #9  
    Forum Isotope Zelos's Avatar
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    Quote Originally Posted by DarcgreY
    Common sence isn't going to tell you that if you double the velocity of an object the kinetic energy is going to be four times. Early scientists like Galileo were suprised when they discovered this relationship through experiments.
    yes but later discoveries were even more supricing that it really isnt a double of speed equals 4x energy but that it increases even faster wich is more noticble at NCV
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    It is common sense, Darcgrey, you just have to see it right. If you whack a baseball and get it going at 20m/s, you've got to run after it and hit it again to get it going at 40m/s. That running after it is a whole pile of extra work. Here's an rehash from ENERGY EXPLAINED that tells you more:

    Consider a 10 kilogram cannonball, in space, travelling at 1000 metres per second. We talk about how much kinetic energy this cannonball has. We talk about KE = ½ MV2 and we do the maths and get five million Joules. But what has the cannonball really got? Its mass seems real enough, and its motion seems real enough too.

    To find out more I take a spacewalk to place a thousand sheets of cardboard in the path of my cannonball. Each sheet of cardboard exerts a small braking force, slowing the cannonball to a halt in two seconds. We know that the cannonball will punch through more cardboard in the first second than in the second second, because it’s slowing down. So we deduce that a cannonball travelling at 1000m/s has more than twice the kinetic energy of one travelling at 500m/s. We can do the arithmetic for each second, then slice the seconds up finer and finer, and we end up realising that the ½V2 is the integral of all the velocities between V and 0.

    The kinetic energy is a way of describing the stopping distance for a given force applied to a given mass moving at a given velocity. You can flip it around to think about force times distance to get something moving. Or you can think in terms of damage. But basically that cannonball has “got” kinetic energy like it has “got” stopping distance.

    It’s similar with momentum. That’s a different way of looking at the mass and the motion, based on force and time instead of force and distance. We look back to our cannonball and cardboard, and we know by definition that in the first second the same amount of time passed as in the second second. So we realise that a cannonball travelling at 1000m/s has twice the momentum of one travelling at 500m/s.

    The momentum is a way of describing the stopping time for a given force applied to a given mass moving at a given velocity. Again you can flip it around, but basically that cannonball has “got” momentum like it has “got” stopping time. Note that momentum is always conserved in a collision because two objects are in contact for the same length of time.
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  12. #11  
    Forum Sophomore DarcgreY's Avatar
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    Quote Originally Posted by Farsight
    So we realise that a cannonball travelling at 1000m/s has twice the momentum of one travelling at 500m/s.
    Isn't the relationship exponential not linear. That's what I was talking about when I said I thought it was counter-intuitive. A cannonball traveling at 1000m/s has four times the momentum of one traveling at 500m/s doesn't it?
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    It's twice the momentum, but four times the kinetic energy.

    Momentum is force times time, so if you stop the 1000m/s cannonball in two seconds with a constant force, in the first second it loses half of its momentum, and in the second second it loses the other half.

    Kinetic energy is force times distance. In the first second its average velocity was 750m/s, while in the second second its average velocity was 250m/s, so it covered more distance in the first second than in the second second. So it lost more Kinetic Energy in the first second than in the second second.
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  14. #13  
    Forum Isotope Zelos's Avatar
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    Momentum is force times time,
    yes, but the defintion is momentum is mass times velocity
    force is defined as delta momentum divided with delta time
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    The definitions are a tad circular, Zelos. And a photon's got momentum but no mass. Well, that's what they say, I say it's got mass, only I'll get into trouble with the definitions there too. The thing is that inertia is like momentum pinned down into one place. See MASS EXPLAINED for more.
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  16. #15  
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    what it is and what defined it is 2 different things
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    This is what I was on about. It's all good stuff:

    http://www.thescienceforum.com/MASS-EXPLAINED-5069t.php

    When we turn our attention from a cannonball to a photon, we have to express the energy and the momentum in a different way. There is no “mass”, so the energy is hf, and the momentum is hf/c. The h here is Planck’s constant of 6.63 x 10-34 Joule-seconds, and is an “action” which is a momentum multiplied by a distance. The f is the frequency per second, and our old friend c is distance over time, which converts a stopping-distance measure into a stopping-time measure. It’s just λ/c or wavelength over frequency, so you can also express the momentum as h/λ. And you can see how that momentum affects a mass via Compton scattering...
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  18. #17  
    Forum Isotope Zelos's Avatar
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    what about it?
    Why photon can have momentum is cause energy and mass is the same so if something have energy it has momentum aswell
    I am zelos. Destroyer of planets, exterminator of life, conquerer of worlds. I have come to rule this uiniverse. And there is nothing u pathetic biengs can do to stop me

    On the eighth day Zelos said: 'Let there be darkness,' and the light was never again seen.

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