The position of a particle is given by where t is in second and r is in meter. (a) Find the of the particle. (b) What is the magnitude and direction of velocity of the particle at t = 2s? - Wired Faculty

Question

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The position of a particle is given by
$space straight r with rightwards arrow on top equals 3 straight t straight i with hat on top minus 2 straight t squared straight j with hat on top plus 4 straight k with hat on top$
where t is in second and r is in meter.
(a) Find the $straight v with rightwards arrow on top space and space straight a with rightwards arrow on top$ of the particle.
(b) What is the magnitude and direction of velocity of the particle at t = 2s?

Solution

(a)
Given,
Position of the particle, $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
∴
Velocity,
$space space straight v with rightwards arrow on top equals straight d over dt left parenthesis straight r with rightwards arrow on top right parenthesis space space space space equals straight d over dt left parenthesis 3 straight t straight i with hat on top minus 2 straight t squared straight j with hat on top plus 4 straight k with hat on top right parenthesis$
and,
Acceleration, $space space space straight a with rightwards arrow on top equals straight d over dt left parenthesis straight v with rightwards arrow on top right parenthesis space equals straight d over dt left parenthesis 3 straight i with hat on top minus 4 straight t straight j with hat on top right parenthesis$
$equals negative 4 straight j with hat on top$
(b) Now,
Velocity, $space space straight v with rightwards arrow on top space equals space 3 straight i with hat on top minus 4 straight t straight j with hat on top$
At time = 2 sec,
$straight v with rightwards arrow on top space equals space 3 straight i with hat on top minus 8 straight j with hat on top$
∴ $open vertical bar straight v with rightwards arrow on top close vertical bar space equals space square root of left parenthesis 3 right parenthesis squared plus left parenthesis 8 right parenthesis squared end root space equals space square root of 74 space equals space 8.54 space straight m divided by straight s$
and $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
At t = 2 sec,
Acceleration, $straight a with rightwards arrow on top space equals space minus 4 straight j with hat on top$
∴ $open vertical bar straight a with rightwards arrow on top close vertical bar space equals space 4$ and directed towards negative direction of X-axis.

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