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I'm solving the following question from Kleppner and Kolenkow:

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My solution differs from the textbook solution not only in the magnitudes in parts (a) and (b) but also the conclusion in part (c). After some analysis, I notice that I took $u$ to be the final velocity of each man relative to the car before jumping, whereas the authors take $u$ to be the final velocity of each man relative to car after jumping.

Clearly, the two situations are physically different, as is evident from the difference in the conclusions of part (c). The impulse imparted by each man upon jumping remains constant and increases respectively as the car gets lighter.

Which interpretation is more likely to happen in a real situation (with identical men of course)? I think is mine, as in that case, the impulse imparted by each man upon jumping remains constant, which makes more intuitive sense as compared to the other interpretation where the impulse imparted decreases for each successive man.

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  • $\begingroup$ I agree that the phrasing is somewhat ambiguous. In my view, there is a clear "before" and a clear "after" in a person jumping off something: before the jump, the relative velocity is of course zero; after the jump, there is some relative velocity $u$ (assuming e.g. a horizontal jump and no subsequent horizontal acceleration in mid-air). In between this "before" and "after" there is some non-constant acceleration of both person and flatcar, that I think it makes sense to disregard. $\endgroup$
    – Marius Ladegård Meyer
    Commented Jun 6 at 6:38
  • $\begingroup$ Also, I think it makes sense that the impulse imparted by each person is not constant (but I have not thought about this for very long...) My reason is that, in the limit of extremely many people the flatcar+people should remain almost unaffected after the jump, whereas in the limit of one person and a flatcar of on-person-mass, they change their speeds by the same amount. $\endgroup$
    – Marius Ladegård Meyer
    Commented Jun 6 at 6:43
  • $\begingroup$ Despite all that, I can still see that your interpretation is very sensible too. $\endgroup$
    – Marius Ladegård Meyer
    Commented Jun 6 at 6:44
  • $\begingroup$ @MariusLadegårdMeyer " My reason is that, in the limit of extremely many people the flatcar+people should remain almost unaffected after the jump, whereas in the limit of one person and a flatcar of on-person-mass, they change their speeds by the same amount." That will hold true even if the impulse is constant. The change in speed is given of the flatcar+people is given by impulse divided by their mass $\endgroup$
    – Vulgar Mechanick
    Commented Jun 6 at 7:03
  • $\begingroup$ @VulgarMechanick Your question about which case being more realistic is not trivial. It concerns how the human body applies force. From my intuition, neither case is realistic. Assuming identical jumping movements, the human muscles apply force over a fixed range of movement i.e. distance. $\endgroup$
    – Vincent Thacker
    Commented Jun 6 at 20:48

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