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Question Number 19200 by Tinkutara last updated on 06/Aug/17
A river of width d is flowing with speed u as shown in the figure. John can swim with maximum speed v relative to the river and can cross it in shortest time T. John starts at A. B is the point directly opposite to A on the other bank of the river. If t be the time John takes to reach the opposite bank, match the situation in the column I to the possibilities in column II. Column I (A) John reaches to the left of B (B) John reaches to the right of B (C) John reaches the point B (D) John drifts along the bank while minimizing the time Column II (p) t = T (q) t > T (r) u < v (s) u > v
$$\mathrm{A}\:\mathrm{river}\:\mathrm{of}\:\mathrm{width}\:{d}\:\mathrm{is}\:\mathrm{flowing}\:\mathrm{with}\:\mathrm{speed} \\ $$$${u}\:\mathrm{as}\:\mathrm{shown}\:\mathrm{in}\:\mathrm{the}\:\mathrm{figure}.\:\mathrm{John}\:\mathrm{can}\:\mathrm{swim} \\ $$$$\mathrm{with}\:\mathrm{maximum}\:\mathrm{speed}\:{v}\:\mathrm{relative}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{river}\:\mathrm{and}\:\mathrm{can}\:\mathrm{cross}\:\mathrm{it}\:\mathrm{in}\:\mathrm{shortest}\:\mathrm{time} \\ $$$${T}.\:\mathrm{John}\:\mathrm{starts}\:\mathrm{at}\:{A}.\:{B}\:\mathrm{is}\:\mathrm{the}\:\mathrm{point} \\ $$$$\mathrm{directly}\:\mathrm{opposite}\:\mathrm{to}\:{A}\:\mathrm{on}\:\mathrm{the}\:\mathrm{other} \\ $$$$\mathrm{bank}\:\mathrm{of}\:\mathrm{the}\:\mathrm{river}.\:\mathrm{If}\:{t}\:\mathrm{be}\:\mathrm{the}\:\mathrm{time}\:\mathrm{John} \\ $$$$\mathrm{takes}\:\mathrm{to}\:\mathrm{reach}\:\mathrm{the}\:\mathrm{opposite}\:\mathrm{bank},\:\mathrm{match} \\ $$$$\mathrm{the}\:\mathrm{situation}\:\mathrm{in}\:\mathrm{the}\:\mathrm{column}\:\mathrm{I}\:\mathrm{to}\:\mathrm{the} \\ $$$$\mathrm{possibilities}\:\mathrm{in}\:\mathrm{column}\:\mathrm{II}. \\ $$$$\boldsymbol{\mathrm{Column}}\:\boldsymbol{\mathrm{I}} \\ $$$$\left(\mathrm{A}\right)\:\mathrm{John}\:\mathrm{reaches}\:\mathrm{to}\:\mathrm{the}\:\mathrm{left}\:\mathrm{of}\:{B} \\ $$$$\left(\mathrm{B}\right)\:\mathrm{John}\:\mathrm{reaches}\:\mathrm{to}\:\mathrm{the}\:\mathrm{right}\:\mathrm{of}\:{B} \\ $$$$\left(\mathrm{C}\right)\:\mathrm{John}\:\mathrm{reaches}\:\mathrm{the}\:\mathrm{point}\:{B} \\ $$$$\left(\mathrm{D}\right)\:\mathrm{John}\:\mathrm{drifts}\:\mathrm{along}\:\mathrm{the}\:\mathrm{bank}\:\mathrm{while} \\ $$$$\mathrm{minimizing}\:\mathrm{the}\:\mathrm{time} \\ $$$$\boldsymbol{\mathrm{Column}}\:\boldsymbol{\mathrm{II}} \\ $$$$\left(\mathrm{p}\right)\:{t}\:=\:{T} \\ $$$$\left(\mathrm{q}\right)\:{t}\:>\:{T} \\ $$$$\left(\mathrm{r}\right)\:{u}\:<\:{v} \\ $$$$\left(\mathrm{s}\right)\:{u}\:>\:{v} \\ $$
Commented by Tinkutara last updated on 06/Aug/17
Commented by ajfour last updated on 07/Aug/17
Commented by ajfour last updated on 07/Aug/17
To minimise the time, θ=0 and john reaches to the right of B. t=T=(d/v) ; drift=((ud)/v) regardless of v>u or v<u . If θ>0 or θ<0 ⇒ t=(d/(vcos θ)) > T If John reaches left of B, θ > sin^(−1) ((u/v)) and v>u , t>T If time is to be minimised, John shouldn′t waste time drfting along the bank. (A)→(q),(r) (B)→(p) or (q), (r) or (s) (C)→(q),(r) (D)→(q),(r) or (s)
$$\mathrm{To}\:\mathrm{minimise}\:\mathrm{the}\:\mathrm{time},\:\theta=\mathrm{0} \\ $$$$\mathrm{and}\:\mathrm{john}\:\mathrm{reaches}\:\mathrm{to}\:\mathrm{the}\:\mathrm{right}\:\mathrm{of}\:\mathrm{B}. \\ $$$$\:\:\:\:\mathrm{t}=\mathrm{T}=\frac{\mathrm{d}}{\mathrm{v}}\:\:;\:\:\mathrm{drift}=\frac{\mathrm{ud}}{\mathrm{v}} \\ $$$$\mathrm{regardless}\:\mathrm{of}\:\mathrm{v}>\mathrm{u}\:\mathrm{or}\:\mathrm{v}<\mathrm{u}\:. \\ $$$$\mathrm{If}\:\theta>\mathrm{0}\:\mathrm{or}\:\theta<\mathrm{0}\:\Rightarrow\:\:\mathrm{t}=\frac{\mathrm{d}}{\mathrm{vcos}\:\theta}\:>\:\mathrm{T} \\ $$$$\mathrm{If}\:\mathrm{John}\:\mathrm{reaches}\:\mathrm{left}\:\mathrm{of}\:\mathrm{B},\: \\ $$$$\:\:\:\:\theta\:>\:\mathrm{sin}^{−\mathrm{1}} \left(\frac{\mathrm{u}}{\mathrm{v}}\right)\:\mathrm{and}\:\mathrm{v}>\mathrm{u}\:,\:\mathrm{t}>\mathrm{T} \\ $$$$\mathrm{If}\:\mathrm{time}\:\mathrm{is}\:\mathrm{to}\:\mathrm{be}\:\mathrm{minimised},\:\mathrm{John} \\ $$$$\mathrm{shouldn}'\mathrm{t}\:\mathrm{waste}\:\mathrm{time}\:\mathrm{drfting} \\ $$$$\mathrm{along}\:\mathrm{the}\:\mathrm{bank}. \\ $$$$\left(\mathrm{A}\right)\rightarrow\left(\mathrm{q}\right),\left(\mathrm{r}\right) \\ $$$$\left(\mathrm{B}\right)\rightarrow\left(\mathrm{p}\right)\:\mathrm{or}\:\left(\mathrm{q}\right),\:\left(\mathrm{r}\right)\:\mathrm{or}\:\left(\mathrm{s}\right) \\ $$$$\left(\mathrm{C}\right)\rightarrow\left(\mathrm{q}\right),\left(\mathrm{r}\right) \\ $$$$\left(\mathrm{D}\right)\rightarrow\left(\mathrm{q}\right),\left(\mathrm{r}\right)\:\mathrm{or}\:\left(\mathrm{s}\right) \\ $$
Commented by Tinkutara last updated on 07/Aug/17
Thank you very much Sir!
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{very}\:\mathrm{much}\:\mathrm{Sir}! \\ $$
Commented by ajfour last updated on 07/Aug/17
thanks.
$$\mathrm{thanks}. \\ $$

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