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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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