Sbe n svkrq fcrrq $z$, gur bcgvzny fbyhgvba sbe gur qhpx vf gb svefg tb gb (gur vafvqr bs) gur pvepyr jvgu enqvhf $1/z$ naq pvepyr guvf hagvy vg unf ernpurq bccbfvgr cunfr bs gur png—juvpu nterrf jvgu lbhe fbyhgvba. Gur erznvavat dhrfgvba vf ubj gb ernpu gur rqtr sebz guvf arj fgnegvat cbfvgvba.
N sevraq bs zvar cbvagrq bhg gung nsgre guvf cbvag gur png jvyy or pvepyvat gur bhgfvqr bs gur pvepyr ng znkvzhz fcrrq va n fvatyr qverpgvba (nf bccbfrq gb gheavat nebhaq rirel abj naq ntnva). Guvf qverpgvba vf qrgrezvarq ol gur genwrpgbel bs gur qhpx vzzrqvngryl nsgre oernxvat $e = 1/z$, ohg nsgre gung vf svkrq fvapr gur nathyne fcrrq bs gur qhpx vf ybjre guna gung bs gur png. Gurersber sbe nal tvira cbvag ba gur havg pvepyr, ertneqyrff bs gur pheir jr pubbfr gb trg gurer, gur png jvyy or geniryvat gb gung cbvag ng znkvzhz fcrrq bire gur ybatre nep (juvpu jr pna rafher ol gnxvat na neovgenel fznyy qrgbhe) bs gur havg pvepyr. Va bgure jbeqf: gur qhpx’f genwrpgbel qbrf abg vasyhrapr gur png nalzber. Gurersber gur bcgvzny cngu gb gnxr sebz urer ba bhg vf fvzcyl gur fubegrfg cngu, v.r. n fgenvtug yvar. Nyy gung erznvaf gb or qrgrezvarq vf juvpu fgenvtug yvar gb gnxr.
Gelvat gb fbyir sbe juvpu natyr vf bcgvzny yrnqf gb n flfgrz bs gjb genaprqragny rdhngvbaf va gjb inevnoyrf (nf vf bsgra gur pnfr jvgu gurfr tbavbzrgevp ceboyrzf), ohg ertneqyrff gurl ner abg pbzcngvoyr jvgu lbhe qrfpevcgvba bs enqvhf naq natyr nf n shapgvba bs gvzr.
I solved this problem as a differential equation. I’ve just realized that the equations above describe a straight line. A tangent to the 1/m circle. The duck moves in a straight line. I found a more complicated way of solving what could have been a simple problem.
V’z abg fher vs jr’er raqvat hc jvgu gur fnzr fgenvtug yvar gubhtu? Znlor zl frg bs genafpraqragny rdhngvbaf unf gur fvzcyr fbyhgvba bs gur gnatrag, juvpu V’ir bireybbxrq, ohg V guvax gur natyr bs gur yvar j.e.g. gur pvepyr bs enqvhf 1/z fubhyq qrcraq ba gur png’f fcrrq.
I disagree. EDIT: I’m no longer sure I disagree
Sbe n svkrq fcrrq $z$, gur bcgvzny fbyhgvba sbe gur qhpx vf gb svefg tb gb (gur vafvqr bs) gur pvepyr jvgu enqvhf $1/z$ naq pvepyr guvf hagvy vg unf ernpurq bccbfvgr cunfr bs gur png—juvpu nterrf jvgu lbhe fbyhgvba. Gur erznvavat dhrfgvba vf ubj gb ernpu gur rqtr sebz guvf arj fgnegvat cbfvgvba.
N sevraq bs zvar cbvagrq bhg gung nsgre guvf cbvag gur png jvyy or pvepyvat gur bhgfvqr bs gur pvepyr ng znkvzhz fcrrq va n fvatyr qverpgvba (nf bccbfrq gb gheavat nebhaq rirel abj naq ntnva). Guvf qverpgvba vf qrgrezvarq ol gur genwrpgbel bs gur qhpx vzzrqvngryl nsgre oernxvat $e = 1/z$, ohg nsgre gung vf svkrq fvapr gur nathyne fcrrq bs gur qhpx vf ybjre guna gung bs gur png. Gurersber sbe nal tvira cbvag ba gur havg pvepyr, ertneqyrff bs gur pheir jr pubbfr gb trg gurer, gur png jvyy or geniryvat gb gung cbvag ng znkvzhz fcrrq bire gur ybatre nep (juvpu jr pna rafher ol gnxvat na neovgenel fznyy qrgbhe) bs gur havg pvepyr. Va bgure jbeqf: gur qhpx’f genwrpgbel qbrf abg vasyhrapr gur png nalzber. Gurersber gur bcgvzny cngu gb gnxr sebz urer ba bhg vf fvzcyl gur fubegrfg cngu, v.r. n fgenvtug yvar. Nyy gung erznvaf gb or qrgrezvarq vf juvpu fgenvtug yvar gb gnxr.
Gelvat gb fbyir sbe juvpu natyr vf bcgvzny yrnqf gb n flfgrz bs gjb genaprqragny rdhngvbaf va gjb inevnoyrf (nf vf bsgra gur pnfr jvgu gurfr tbavbzrgevp ceboyrzf), ohg ertneqyrff gurl ner abg pbzcngvoyr jvgu lbhe qrfpevcgvba bs enqvhf naq natyr nf n shapgvba bs gvzr.
I solved this problem as a differential equation. I’ve just realized that the equations above describe a straight line. A tangent to the 1/m circle. The duck moves in a straight line. I found a more complicated way of solving what could have been a simple problem.
V’z abg fher vs jr’er raqvat hc jvgu gur fnzr fgenvtug yvar gubhtu? Znlor zl frg bs genafpraqragny rdhngvbaf unf gur fvzcyr fbyhgvba bs gur gnatrag, juvpu V’ir bireybbxrq, ohg V guvax gur natyr bs gur yvar j.e.g. gur pvepyr bs enqvhf 1/z fubhyq qrcraq ba gur png’f fcrrq.