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Question Number 70035 by TawaTawa last updated on 30/Sep/19

Commented by TawaTawa last updated on 30/Sep/19

Commented by mr W last updated on 30/Sep/19

$$letx=XP$$$$AP=\sqrt{a^2+x^2}$$$$PY=b-x$$$$t=\frac{AP}{v}+\frac{PY}{u}$$$$uvt=u\sqrt{a^2+x^2}+v(b-x)$$$$\frac{d\left(uvt\right)}{dx}=\frac{ux}{\sqrt{a^2+x^2}}-v=0$$$$\Rightarrow ux=v\sqrt{a^2+x^2}$$$$u^2{x}^2=v^2(a^2+x^2)$$$$x^2=\frac{v^2{a}^2}{u^2-v^2}$$$$x=\frac{a}{\sqrt{\frac{u^2}{v^2}-1}}$$$$t_{min}=\frac{\sqrt{a^2+x^2}}{v}+\frac{b-x}{u}=\frac{1}{u}[(\frac{u^2}{v^2}-1)x+b]$$$$\Rightarrow t_{min}=\frac{1}{u}[a\sqrt{\frac{u^2}{v^2}-1}+b]$$

Commented by TawaTawa last updated on 30/Sep/19

$$\mathrm{Wow},\mathrm{God}\mathrm{bless}\mathrm{you}\mathrm{sir}.\mathrm{I}\mathrm{appreciate}.$$

Commented by TawaTawa last updated on 30/Sep/19

$$\mathrm{Sir},\mathrm{help}\mathrm{me}\mathrm{check}\mathrm{question}70048$$

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