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Motion In A Plane

Question
CBSEENPH11018559

Show that there are two angles of projection for which horizontal range is the same.

Solution

Horizontal range is given by,
                   straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction
where, straight theta is the angle of projection and is less than 90 degree.
We know, sin left parenthesis 180 degree minus 2 straight theta right parenthesis space equals space sin space 2 straight theta
rightwards double arrow             sin space 2 left parenthesis 90 degree minus straight theta right parenthesis space equals space sin space 2 straight theta
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#6 {main}</pre>         [Where straight alpha equals 90 degree minus straight theta]
Now,  
         straight R subscript 0 equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction  and  straight R subscript straight alpha space equals space fraction numerator straight u squared space sin 2 straight alpha over denominator straight g end fraction
Since     sin space 2 straight theta space equals space sin space 2 straight alpha
∴            straight R subscript 0 equals straight R subscript straight alpha
Thus horizontal range is the same for angle of projection θ and α = 90°–θ.