## Minimum Time to Move Between Fixed Points Through a Point Moving Along x Axis

A point P is free to move along the\[x\]

axis between points A and B.Line segments joint to point P to points R and S as shown, and the lines PR and PS make angles \[\theta\]

and \[\phi\]

with the positive and negative \[y\]

axes respectively. How can you find the minimum time to move from R to Q?\[tan \theta =\frac{AB-x}{b} \rightarrow sec^2 \theta \frac{d \theta}{dx}=- \frac{1}{b} \rightarrow \frac{d \theta}{dx}=- \frac{cos^2 \theta}{b}\]

\[tan \phi =\frac{x}{a} \rightarrow sec^2 \phi \frac{d \phi}{dx}= \frac{1}{a} \rightarrow \frac{d \phi}{dx}= \frac{cos^2 \phi}{a}\]

Then

\[\frac{d \theta}{d \phi}=\frac{\frac{d \theta}{dx}}{\frac{d \phi}{dx}}=\frac{- \frac{cos^2 \theta}{b}}{ \frac{cos^2 \phi}{a}}=- \frac{a cos^2 \theta}{b cos^2 \phi}\]

Let a particle move from R to P with speed

\[v_1\]

and from P to Q with speed \[v_2\]

.The time taken to move from R to P is

\[t_1=\frac{b}{v_1 cos \theta}\]

.The time taken to move from P to Q is

\[t_2=\frac{a}{v_2 cos \phi}\]

.The time taken to move from R to Q is then

\[T=\frac{b}{v_1 cos \theta}+ \frac{a}{v_2 cos \phi}\]

.\[\begin{equation} \begin{aligned} \frac{dT}{d \theta}&= \frac{b sec \theta tan \theta}{v_1 } + \frac{a}{v_2 sec \phi tan \phi} \frac{d \phi}{d \theta} \\ &= \frac{b sec \theta tan \theta}{v_1 } + \frac{a}{v_2 sec \phi tan \phi} (- \frac{b cos^2 \phi}{a cos^2 \theta}) \\ &= b sec^2 \theta (\frac{sin \theta}{v_1}-\frac{sin \phi}{v_2}) =0 \end{aligned} \end{equation}\]

.Hence

\[\frac{sin \theta}{v_1}-\frac{sin \phi}{v_2} =0 \rightarrow \frac{sin \theta}{sin \phi}=\frac{v_1}{v_2}\]

is a condition for what is obviously a minimum time.