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In the textbook Fundamentals of Physics by R. Shankar of Yale Open Courses, appears the following assertion pertaining to a car driving off a cliff, which seems correct:

This is exactly how long it would take to hit the ground had it simply toppled over the edge from rest. The horizontal velocity does not delay the crash one bit (unless you take into account the curvature of the earth).

However, at a later point in the book, Shankar makes the following claim about a bullet fired horizontally from a gun:

The bullet hits the ground at point 1, under the pull of gravity. If you fire another bullet at a greater velocity it will land a little further away, at point 2. While greater initial speed will extend the time of flight even on a flat earth, the flight is further enhanced by the earth curving under the bullet.

The claim that "While greater initial speed will extend the time of flight even on a flat earth..." seems to contradict the the initial claim.

Is this a blatant error on the part of the textbook?

Here is a link to the book if anybody is curious.

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    $\begingroup$ This indeed seems to be an error on the author's part. If the earth is flat and we assume a constant acceleration under gravity, the time an object takes to hit the ground must only depend on the initial VERTICAL velocity. This is because the vertical motion is independent of the horizontal motion and the only force on the object is in the vertical direction. Furthermore, even if the acceleration under gravity is taken to vary with height , this will still be true. I think the author meant that higher initial horizontal velocity enhances the length of the flight and not the time. $\endgroup$ Commented Jul 8 at 9:40
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    $\begingroup$ I think that it is a typo - While greater initial speed will extend the distance of flight even on a flat earth. $\endgroup$
    – Farcher
    Commented Jul 8 at 14:08
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    $\begingroup$ @RatulThakur But why did you post this as a comment instead of an answer? $\endgroup$ Commented Jul 8 at 20:38
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    $\begingroup$ I took the liberty to email Professor Shankar, who has confirmed the erratum below. :) $\endgroup$
    – CR Drost
    Commented Jul 8 at 21:56
  • $\begingroup$ Now also confirmed in the errata campuspress.yale.edu/rshankar/files/2024/07/… $\endgroup$ Commented Jul 10 at 6:19

2 Answers 2

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I agree that the statement I made in the book is wrong: On a flat earth the time of flight is not affected by the initial horizontal velocity, only the horizontal distance is. I will post a correction in the errata in the website. Thank you for bringing this to my attention. R.Shankar Yale

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As @RatulThakur points out, this is pretty clearly an error. ToF is not affected at all by horizontal velocity on a flat Earth, unless you're considering the aerodynamics of the object (there's a reason why planes need to stay at a certain speed).

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  • $\begingroup$ *ToF on a flat earth. On a round earth (or even more arbitrary shapes) "horizontal" velocity does of course have an effect $\endgroup$
    – Tristan
    Commented Jul 9 at 9:31
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    $\begingroup$ @Tristan The book already mentions that adjustment in both places. $\endgroup$
    – Barmar
    Commented Jul 9 at 14:20
  • $\begingroup$ @Barmar yes, but the answer makes an incorrect claim ("ToF is not affected at all by horizontal velocity, unless you're considering the aerodynamics of the object") by not also including that requirement $\endgroup$
    – Tristan
    Commented Jul 9 at 14:44
  • $\begingroup$ @Tristan Because they're only addressing the error in the book, not the correct part about curvature. $\endgroup$
    – Barmar
    Commented Jul 9 at 14:46
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    $\begingroup$ @Tristan Sufficient velocity makes the ToF infinite :-). $\endgroup$ Commented Jul 11 at 9:50

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