CBSE Class 9 Answered

An object starts linear motion with a velocity 'u' and under uniform acceleration 'a' it acquires a velocity 'v' in time 't'. Draw its velocity time graph. From this graph obtain the following equations. (a) v = u + at (b) s = ut + ½ at²

Asked by Topperlearning User | 16 Jun, 2016, 05:09: PM

Expert Answer

$begin mathsize 12px style At space straight t space equals space 0 comma space initial space velocity space is space straight u space and space then space increases space to space straight v space left parenthesis at space point space straight B right parenthesis space in space time space straight t. space space space left parenthesis straight a right parenthesis space Draw space straight a space perpendicular space line space BC space and space BE space from space point space straight B space on space time space and space the space velocity space axis space respectively comma space so space that space OA space equals space straight u comma space OE space equals space BC space equals space straight v. space Also space draw space straight a space line space AD space parallel space to space time space axis comma space so space that space OC space equals space AD space equals space straight t space Then space change space in space velocity space in space time space straight t space is space BC space minus space OA space equals space BC space minus space CD space equals space BD space But space BC space equals space straight v space and space CD space equals space OA space equals space straight u space Hence comma space BD space equals space straight v space minus space straight u space From space the space velocity space minus space time space graph comma space the space acceleration space of space the space object space is space given space by straight a equals fraction numerator change space in space velocity over denominator time space taken end fraction equals BD over AD equals BD over OC equals fraction numerator straight v minus straight u over denominator straight t end fraction therefore space at equals straight v minus straight u space space space space space space space straight v equals straight u plus at left parenthesis straight b right parenthesis space According space to space velocity space minus space time space graph comma space the space total space distance space covered space by space the space object space is space obtained space by space the space area space under space the space graph. space Hence comma space the space distance space apostrophe straight s apostrophe space travelled space by space the space object space space space space space equals space Area space of space the space trapezium space OABC space space space space space equals space Area space of space rectangle space OADC space plus space Area space of space ADB space space space space equals OA cross times OC plus 1 half left parenthesis AD cross times BD right parenthesis space space space space space But space OA equals straight u comma space OC equals straight t comma space therefore AD equals OC equals straight t BD space equals space change space in space velocity space equals space straight u space minus space straight v space equals space at space space space space straight s equals ut plus 1 half left parenthesis at cross times straight t right parenthesis space space space space straight s equals ut plus 1 half at squared end style$