Prove that the maximum horizontal range is 4 times the maximum height attained by a projectile. from Physics Motion in A Plane Class 11 Manipur Board
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Prove that the maximum horizontal range is 4 times the maximum height attained by a projectile.

The horizontal range is maximum when the angle of projection is 45°.

So,

space space space space straight R equals fraction numerator straight u squared sin 2 straight theta over denominator straight g end fraction equals fraction numerator straight u squared sin left parenthesis 2 cross times 45 degree right parenthesis over denominator straight g end fraction equals straight u squared over straight g        ...(1)

Maximum height for angle of projection 45 degree is,

space space space straight H equals fraction numerator straight u squared sin squared straight theta over denominator 2 straight g end fraction equals fraction numerator straight u squared sin squared 45 degree over denominator 2 straight g end fraction equals fraction numerator straight u squared over denominator 4 straight g end fraction              ...(2) 

Therefore, from equation (1) and (2),

                      R = 4H

That is, range is 4 times the maximum height attained by a projectile. 

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