A ball is projected at angle 0 from a point O ↔ (0,0). After passing the ball through a point (x, y) it reaches the ground at a distance R from O. Show that from Physics Motion in A Plane Class 11 Manipur Board
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Motion in A Plane

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Physics Part I

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Physics

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A ball is projected at angle 0 from a point O ↔ (0,0). After passing the ball through a point (x, y) it reaches the ground at a distance R from O. Show that
straight y over straight x equals tan space straight theta open parentheses 1 minus straight x over straight R close parentheses

Let the ball be projected with speed u.

The equation of trajectory is,

         straight y equals straight x space tanθ space minus space 1 half straight g fraction numerator straight x squared over denominator straight u squared cos squared straight theta end fraction

On taking x tanstraight theta common, we get

or      straight y equals straight x space tanθ open parentheses 1 minus 1 half straight g fraction numerator straight x over denominator straight u squared cos squared straight theta space tanθ end fraction close parentheses 

or     space straight y equals straight x space tanθ open parentheses 1 minus straight g fraction numerator straight x over denominator 2 straight u squared space cosθ space sinθ end fraction close parentheses

or     space space straight y equals straight x space tanθ open parentheses 1 minus fraction numerator straight x over denominator open parentheses begin display style bevelled fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction end style close parentheses end fraction close parentheses

or      straight y over straight x equals tanθ open parentheses 1 minus straight x over straight R close parentheses

Hence, proved the result. 
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