A particle moves in XY plane, such that its position at any instant is and Find the angle between velocity vector and acceleration vector at any instant. - Wired Faculty

Question

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A particle moves in XY plane, such that its position at any instant is
$straight x equals straight alpha space sin space ωt$ and $straight y equals straight alpha left parenthesis 1 minus cos space ωt right parenthesis$
Find the angle between velocity vector and acceleration vector at any instant.

Solution

The position of the particle at any instant is,
$<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
Therefore the velocity of particle at any instant is,
 $straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis space equals space straight d over dt open parentheses αsinωt straight i with hat on top plus straight alpha left parenthesis 1 minus cosωt right parenthesis straight j with hat on top close parentheses$
$rightwards double arrow$ $space straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis space equals space αω space cosωt straight i with hat on top plus αω space sinωt straight j with hat on top$
Acceleration of particle at any instant is,
 $straight a with rightwards arrow on top equals straight d over dt left parenthesis straight v with rightwards arrow on top right parenthesis equals straight d over dt open parentheses aω space cosωt space straight i with hat on top plus space αωsinωt space straight j with hat on top close parentheses$
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Now,
$straight a with rightwards arrow on top times straight v with rightwards arrow on top equals left parenthesis negative αω squared sinωt straight i with hat on top plus αω squared cosωt straight j with hat on top right parenthesis times left parenthesis αω space cosωt space straight i with hat on top plus αωsinωt space straight j with hat on top right parenthesis$
$equals negative straight alpha squared straight omega cubed sinωtcosωt plus straight alpha squared straight omega cubed cosωtsinωt equals 0$
Since dot product of acceleration vector and velocity vector is zero, therefore the acceleration vector and velocity vector is $90 degree.$

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Motion In A Plane

A particle moves in XY plane, such that its position at any instant is
$straight x equals straight alpha space sin space ωt$ and $straight y equals straight alpha left parenthesis 1 minus cos space ωt right parenthesis$
Find the angle between velocity vector and acceleration vector at any instant.

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