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I Velocity addition and conservation of the energy

  1. Dec 4, 2017 #1
    Hello everyone,

    For some time I'm a little bit confused about (at the first view) a very simple question, which is about the conversation of the energy of moving objects (in terms of special relativity).

    As an example lets talk about firearms. If the mass of the gun M1 is infinitely higher than the mass of the bullet M2, then all the kinetic energy of the shot will be imparted to the bullet. Now, lets take the case of the moving gun-bullet system (in X-direction). In the frame of the static observer (S) the total velocity of the bullet (or kinetic energy) will be different as a function of the direction of the shot (transverse or longitudinal); since sqrt(Vx2 + Vy2) < Vx + Vy.

    As far as I understand the total kinetic energy of the system is independent of the direction of the shot, then how one can explain the total energy difference between transverse and longitudinal shot? My first guess was that the answer was hiding in the distribution of the energy between the gun and the bullet; however I estimate that this guess is wrong since M1 (gun) >>>>> M2 (bullet) ->> all the energy of the shot is imparted to the bullet.
     
  2. jcsd
  3. Dec 4, 2017 #2

    Ibix

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    Your first guess is correct. If ##M_1## is infinite, then in any frame where the gun is moving the kinetic energy is infinite both before and after the shot is fired. You are unlikely to get sensible results.

    Try again with a finite mass for the gun.
     
  4. Dec 4, 2017 #3

    Let's first forget all relativistic complications.

    Non-moving gun-bullet system:

    If the mass ratio is x, then the ratio of accelerations is x, ratio of final speeds is x, ratio of distances traveled is x2, ratio of energies gained or lost is x2


    Moving gun-bullet system:

    Energy of the shot is imparted mostly to the object that moves the most.

    Ratio of energies = ratio of distances traveled
     
  5. Dec 4, 2017 #4
    Actually ratio of distances traveled is x, not x2

    And also ratio of energies gained or lost is x, not x2

    You see, this basic formula holds:
    E=F*d
     
  6. Dec 5, 2017 at 7:16 AM #5
    Thank you for the responses. Indeed, problem was coming from "infinite" mass assumption.
     
  7. Dec 11, 2017 at 4:29 AM #6
    Hello again,

    I would like to ask a more practical question about the energy conservation.

    FIRST SCENARIO: Again let's take as an example a gunshot. After the shot, bullet attains the speed Vy = 0.7c in the frame S (approximation = almost all the energy of the shot goes to the bullet).
    The kinetic energy of the bullet can be calculated though the formula: KE = mc2 - m0c2; if we put c = 1 and m0= 1; then KE = mVy - 1.
    Here, KE = 1/sqrt(1 - Vy2) - 1 = 0.4003.


    SECOND SCENARIO: Now the exact same replica of the system is moving with the speed Vr = 0.9c in the frame S. The same shot is realized in the transverse direction. The velocity of the bullet is Vy=0.7c in the S' frame and Vy' in the S frame. The transverse velocity transformation formula is Vy' = sqrt(1 - Vr2)*Vy. That gives us Vy' = 0.3051. After the shot, the total velocity of the bullet in S frame equals to: Vtot = sqrt(Vy'2 + Vr2) = 0.9503.
    The KE of the bullet gained from the shot equals to: KE = mVtot - mVr = 0.9183.

    QUESTION:
    I expected that the kinetic energy of the bullet gained from the shot would be equal in both scenarios because of the conservation of the energy. Is there any mistake in my calculations or in my reasoning?

    Thank you
     
    Last edited: Dec 11, 2017 at 7:56 AM
  8. Dec 11, 2017 at 6:07 AM #7

    Mister T

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    You've given an example of the same scenario analyzed in two different frames. When a quantity is the same in two different frames we say it's invariant. On the other hand, when a quantity stays the same in one frame we say it's conserved.
     
  9. Dec 11, 2017 at 7:22 AM #8
    I'm a little bit confused... If we analyze this situation only from the point of view of the observer in the frame (S). In one hand we have a gun which doesn't' move and in the over hand an equal gun which is moving with the speed Vr within S.

    If the observer wants to measure KE of the bullet gained from the shot he should find the same energy for both stationary and moving systems (because of the conservation of energy)?
     
  10. Dec 11, 2017 at 7:28 AM #9

    jbriggs444

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    What about the work done on the gun/planet by the bullet? If you fire a shot in the forward direction, this quantity is non-zero.
     
  11. Dec 11, 2017 at 7:31 AM #10
    I thank you for your reply. That's why I restrained the problem only to the transverse shot, in order to simplify the calculations.
     
  12. Dec 11, 2017 at 8:09 AM #11
    The transverse shot is not really a transverse shot, because the longitudinal momentum of the bullet increases, as the gunpowder gives some of its longitudinal momentum to the bullet. The gunpowder has less mass and less longitudinal momentum after the firing.
     
  13. Dec 11, 2017 at 10:07 AM #12

    PeroK

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    If you have the bullet fired in the ##y## direction at speed ##v_y##, then the gain in KE is

    ##(\gamma_y -1)mc^2##.

    If you analyse this in a frame where the experiment is moving in the ##x## direction at speed ##v_x## then the gain in KE is:

    ##\gamma_x (\gamma_y -1)mc^2##

    The two coincide, therefore, only when ##v_x## in non relativistic.

    Note: It might be a useful exercise to derive the above expressions.
     
  14. Dec 11, 2017 at 10:29 AM #13

    PAllen

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    hmm. I get product of gammas -1, rather than with 1 in parentheses as you have:

    ##(\gamma_x \gamma_y -1)mc^2##
     
  15. Dec 11, 2017 at 10:56 AM #14

    PeroK

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    I got ##\gamma' = \gamma_x \gamma_y## for the gamma factor in the new frame.

    In that frame the initial energy is ##\gamma_x mc^2##.
     
  16. Dec 11, 2017 at 11:08 AM #15

    PAllen

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    Ah, you want increase in KE, not total KE. I was computing total KE.

    (fyi, if you don't know it, deriving using 4 vectors the general case is trivial:

    ##\gamma' = \gamma_u \gamma_v## (1-u⋅v)

    where bolded are 3 vectors.
    )
     
    Last edited: Dec 11, 2017 at 11:36 AM
  17. Dec 11, 2017 at 11:33 AM #16

    Mister T

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    The conservation law tells you that something you determine using one frame of reference will have the same value both before and after a process. It does not tell you that the value of something will be the same when determined in one frame of reference as it is when determined in another frame of reference.
     
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