Deprecated: Methods with the same name as their class will not be constructors in a future version of PHP; plgContentJComments has a deprecated constructor in /var/www/astarmathsandphysics/plugins/content/jcomments/jcomments.php on line 25 Call Stack: 0.0001 361232 1. {main}() /var/www/astarmathsandphysics/index.php:0 0.1389 1210072 2. Joomla\CMS\Application\SiteApplication->execute() /var/www/astarmathsandphysics/index.php:49 0.1389 1210072 3. Joomla\CMS\Application\SiteApplication->doExecute() /var/www/astarmathsandphysics/libraries/src/Application/CMSApplication.php:267 0.2319 4190592 4. Joomla\CMS\Application\SiteApplication->dispatch() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:233 0.2332 4218312 5. Joomla\CMS\Component\ComponentHelper::renderComponent() /var/www/astarmathsandphysics/libraries/src/Application/SiteApplication.php:194 0.2340 4236024 6. Joomla\CMS\Component\ComponentHelper::executeComponent() /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:356 0.2342 4266544 7. require_once('/var/www/astarmathsandphysics/components/com_content/content.php') /var/www/astarmathsandphysics/libraries/src/Component/ComponentHelper.php:381 0.2353 4289264 8. ContentController->execute() /var/www/astarmathsandphysics/components/com_content/content.php:42 0.2353 4289264 9. ContentController->display() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:710 0.2888 4961528 10. ContentController->display() /var/www/astarmathsandphysics/components/com_content/controller.php:113 0.2925 5153680 11. Joomla\CMS\Cache\Controller\ViewController->get() /var/www/astarmathsandphysics/libraries/src/MVC/Controller/BaseController.php:663 0.2931 5174608 12. ContentViewArticle->display() /var/www/astarmathsandphysics/libraries/src/Cache/Controller/ViewController.php:102 0.3056 5370400 13. Joomla\CMS\Plugin\PluginHelper::importPlugin() /var/www/astarmathsandphysics/components/com_content/views/article/view.html.php:189 0.3056 5370656 14. Joomla\CMS\Plugin\PluginHelper::import() /var/www/astarmathsandphysics/libraries/src/Plugin/PluginHelper.php:182

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

A point P is free to move along the
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
with the positive and negative
axes respectively. How can you find the minimum time to move from R to Q?

rate of change problem

\[tan \theta =\frac{AB-x}{b} \rightarrow sec^2 \theta \frac{d \theta}{dx}=- \frac{1}{b} \rightarrow \frac{d \theta}{dx}=- \frac{cos^2 \theta}{b}\]

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

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

Let a particle move from R to P with speed  
  and from P to Q with speed  
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  
\[T=\frac{b}{v_1 cos \theta}+ \frac{a}{v_2 cos \phi}\]
\[\begin{equation} \begin{aligned} \frac{dT}{d \theta}&= \frac{b sec \theta tan \theta}{v_1 } + \frac{a}{v_2 sec \phi tan \phi} \frac{d \phi}{d \theta} \\ &= \frac{b sec \theta tan \theta}{v_1 } + \frac{a}{v_2 sec \phi tan \phi} (- \frac{b cos^2 \phi}{a cos^2 \theta}) \\ &= b sec^2 \theta (\frac{sin \theta}{v_1}-\frac{sin \phi}{v_2}) =0 \end{aligned} \end{equation}\]
\[\frac{sin \theta}{v_1}-\frac{sin \phi}{v_2} =0 \rightarrow \frac{sin \theta}{sin \phi}=\frac{v_1}{v_2}\]
  is a condition for what is obviously a minimum time.

Add comment

Security code