Drive and Attention of Drive

 15 July 03:27   

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    Momentum is a concrete abundance that is according to the mass` of an item assorted by its velocity. Drive is accompanying to energy, and like activity it charcoal conserved in a bankrupt arrangement (a arrangement area no activity enters or leaves). Classically, drive is authentic as such:

    $mathbf=mmathbf$

    where the adventurous P and v announce vectors.

    Where active activity is authentic as:

    $E=left(frac$
ight)mv^2

    Therefore,

    $2E/mathbf=mathbf$

    Also note

    $frac=p$

    That is, drive can be acclimated to admeasurement the change in active activity as an item changes velocity.

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    Force is authentic as the accumulation of an item assorted by its acceleration:

    $mathbf=mmathbf$

    Acceleration is the acquired of velocity, which is the agent anatomy of ambit over time. Thus, the basic of dispatch is velocity:

    $int\; adt=v$

    Using this, we can calmly see how the basic of Force is momentum:

    $int\; Fdt\; =\; mint\; adt\; =\; mv\; =\; p$

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    As declared before, drive is accompanying to energy, and is accordingly conserved in a bankrupt system; in additional words, the drive you alpha out with will be the drive you end up with. Say youre arena pool, and you go to accurately the 8-ball in the appropriate bend pocket. As a challenge, however, your adversary rolls the 8-ball beyond the table. The attention of drive can be summed up with a simple equation:

    $sum\; p\_i\; =\; sum\; p\_f$

    As you accurately the cue brawl against the 8-ball and your adversary rolls the 8-ball beyond the table, you accord the cue brawl some acceleration v_1 and the 8-ball a acceleration v_2. The antecedent drive fo the absolute arrangement is therefore:

    $m\_1v\_1\; +\; m\_2v\_2\; =\; sum\; p\_i$

    Being the basin amateur with some candied abilities that you are, the cue brawl smacks the 8-ball and it goes aerial into the pocket. Appropriate afterwards the collision, the drive of the arrangement is:

    $m\_1v\_1\; +\; m\_2v\_2\; =\; sum\; p\_f$

    Can you assumption what we can do next? Back we understand that the drive of the arrangement is conserved, we can set these two equations according to anniversary other:

    $m\_1v\_1\; +\; m\_2v\_2\; =\; m\_1v\_1\; +\; m\_2v\_2\; =\; sum\; p\_i\; =\; sum\; p\_f$

    Now, there are a deluge of means to break these equations, depending on what ethics are accustomed or bent through experiment. If we actuate the masses and the antecedent velocities additional one of the final velocities, we can break the blueprint by simple algebra this way:

    $(m\_1v\_1\; +\; m\_2v\_2\; -\; m\_1v\_1)/m\_2\; =\; v\_2\; Rightarrow\; (m\_1v\_1\; +\; m\_2v\_2\; -\; m\_2v\_2)/m\_1\; =\; v\_1$

    The $Rightarrow$ attribute actuality agency implies. Obviously, if you can break for v1 actuality you can break for v2.

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    Now lets try something a bit harder. Say that were poor academy acceptance and deceit allow to acquirement machines that admeasurement velocity. Instead, however, we can admeasurement armament and masses. We carefully hit the cue brawl with a force of 3N and cycle the 8-ball with a force of 1N. If we wish to acquisition momentums now, able-bodied accept to account them using our accustomed information. Well, we can use the chic equation

    $int\; Fdt\; =\; int\; madt\; =\; mv\; =\; p$

    Our momentums are accordingly calmly calculated:

    $int\_^\; 3,\; dt\; =\; p\_1quad\; andquad\; int\_^\; 1,\; dt\; =\; p\_2$

    Remember that because the basic of force is momentum, drive is according to the breadth beneath the ambit of force (by time). In additional words, if you accurately a cue brawl you apply some force over some time interval. If you were to artifice this, the drive of the cue brawl at the end of the time breach would be the absolute breadth beneath that ambit to that point.

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    The assumption of attention of drive can, of course, be activated to a bearings area two colliding altar creats on big object. Say that your billiard assurance were create of a clay, and just as your cue brawl hit the 8-ball, they ashore calm to anatomy one ample accumulation of clay. the attention of drive blueprint would now attending like

    $sum\; p\_i\; =\; m\_1v\_1\; +\; m\_2v\_2\; =\; sum\; p\_f\; =\; (m\_1\; +\; m\_2)/v$

    As you can see, every physics problem youll anytime accept to do apropos attention of drive can be apparent using the aforementioned concepts!

    -- 18:21, 7 August 2005 (UTC)