Question

Two cars A and B are travelling in same direction with speed $straight v subscript 1 space and space straight v subscript 2$ respectively $left parenthesis straight v subscript 1 greater than straight v subscript 2 right parenthesis.$ When car A is d distance behind the car B, the brakes on A are applied, causing retardation $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$ Show that to prevent a collision between A and B it is necessary that
$left parenthesis straight v subscript 1 minus straight v subscript 2 right parenthesis squared less than left parenthesis 2 αd right parenthesis$

Solution

Here, inorder to prevent the collision of car A with B, the relative velocity of car A w.r.t. B should reduce to zero before car A undergoes the relative displacement of d w.r.t. car B.
Initial velocity of car A =

straight v subscript 1

Initial velocity of car B =

straight v subscript 2

Therefore,
Initial relative velocity of car A w.r.t. B is,

straight v equals straight v subscript 1 minus straight v subscript 2

Acceleration of car A =

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Acceleration of car B = 0
Acceleration of car A w.r.t. car B =

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The relative displacement of car A w.r.t car B, when the relative velocity of car A w.r.t. B decreases to zero is given by,

straight S equals fraction numerator 0 squared minus straight v squared over denominator negative 2 straight alpha end fraction equals fraction numerator straight v squared over denominator 2 straight alpha end fraction equals fraction numerator left parenthesis straight v subscript 1 minus straight v subscript 2 right parenthesis squared over denominator 2 straight alpha end fraction

For collision to avert S should be less than d.
i.e. S < d

rightwards double arrow

fraction numerator left parenthesis straight v subscript 1 minus straight v subscript 2 right parenthesis squared over denominator 2 straight alpha end fraction less than straight d

rightwards double arrow

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