CBSE Class 11-science Answered

A particle is projected from ground at an angle of 60° with horizontal at a speed of 50m/s . The ratio of radius of curvature of trajectory at maximum height to at the point of projection is ?

Asked by arunavamitra50 | 07 Jul, 2018, 08:26: PM

Expert Answer

Parabolic equation of projectile :-

begin mathsize 12px style y space equals space x space tan alpha space minus space fraction numerator g over denominator 2 u squared cos squared alpha end fraction space x squared end style

where u is initial projection velocity and α is angle of projection

Radius of curvature,

begin mathsize 12px style R space equals space fraction numerator open parentheses 1 plus open parentheses begin display style fraction numerator d y over denominator d x end fraction end style close parentheses squared close parentheses to the power of begin display style bevelled 3 over 2 end style end exponent over denominator open vertical bar begin display style fraction numerator d squared y over denominator d x squared end fraction end style close vertical bar end fraction end style

begin mathsize 12px style fraction numerator d y over denominator d x end fraction space equals space tan space alpha space minus space fraction numerator g over denominator u squared cos squared alpha end fraction space x space space semicolon space fraction numerator d squared y over denominator d x squared end fraction space equals space minus space fraction numerator begin display style g end style over denominator begin display style u squared cos squared alpha end style end fraction end style

at projection point :-

begin mathsize 12px style fraction numerator d y over denominator d x end fraction space equals space tan alpha space semicolon space space space space space space R space equals space fraction numerator begin display style open parentheses 1 plus tan squared alpha close parentheses to the power of bevelled 3 over 2 end exponent end style over denominator begin display style bevelled fraction numerator g over denominator open parentheses u squared cos squared alpha close parentheses end fraction end style end fraction space equals space u squared over g cos alpha end style

at the point of maximum height :-

begin mathsize 12px style fraction numerator d y over denominator d x end fraction space equals space 0 semicolon space space R space equals space fraction numerator u squared cos squared alpha over denominator g end fraction end style

it can be seen that required ratio as per question is cosα

Answered by Thiyagarajan K | 08 Jul, 2018, 09:32: AM

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