If 'R' is the horizontal range and 'H' is the maximum height achieved by the projectile at an angle 'ϑ'.  Show that maximum range = (R^2/8H)+(2H)

Asked by Mohit RaAz | 6th Aug, 2015, 08:19: PM

Expert Answer:

begin mathsize 14px style Let space straight u space be space the space velocity space of space the space projection space and space straight theta space be space the space angle space of space projection. Range comma space straight R equals straight u squared over straight g sin space 2 straight theta space space space space space space space space space space space space space space space space space space space space space space space space... space left parenthesis 1 right parenthesis Height comma space straight h equals fraction numerator straight u squared space sin squared space straight theta over denominator 2 space straight g end fraction space space space space space space space space space space space space space space space space space space space space space space space space... space left parenthesis 2 right parenthesis From space the space equation space left parenthesis 1 right parenthesis space and space left parenthesis 2 right parenthesis comma space we space get comma straight h over straight R equals fraction numerator straight u squared space sin squared space straight theta over denominator 2 space straight g end fraction cross times fraction numerator straight g over denominator straight u squared space sin space 2 straight theta end fraction straight h over straight R equals fraction numerator sin squared space straight theta over denominator 2 space sin space 2 straight theta end fraction equals fraction numerator sin squared straight theta over denominator 2 left parenthesis 2 sin space straight theta space cos space straight theta right parenthesis space end fraction equals 1 fourth tan space straight theta space space space space space space space left square bracket because sin space 2 straight theta equals 2 sin space straight theta space cos space straight theta right square bracket tan space straight theta equals fraction numerator 4 straight h over denominator straight R end fraction end style
begin mathsize 14px style tan space straight theta equals fraction numerator opposite space side over denominator adjacent space side end fraction end style
 
begin mathsize 14px style straight x equals square root of straight R squared plus left parenthesis 4 straight h right parenthesis squared end root equals square root of straight R squared plus 16 straight h squared end root sin space straight theta equals fraction numerator opposite space side over denominator hypotenuse end fraction equals fraction numerator 4 straight h over denominator straight x end fraction equals fraction numerator 4 straight h over denominator square root of straight R squared plus 16 straight h squared end root end fraction space space space space... space left parenthesis 3 right parenthesis space Maximum space range comma space Rmax equals straight u squared over straight g space space space space space left square bracket From space eqn. space left parenthesis 1 right parenthesis right square bracket From space eqns. space left parenthesis 2 right parenthesis comma space we space get straight u squared over straight g equals fraction numerator 2 straight h over denominator sin squared straight theta end fraction space space space space space space... space left parenthesis 4 right parenthesis therefore From space eqns. space left parenthesis 3 right parenthesis space and space left parenthesis 4 right parenthesis comma space we space get straight R subscript max equals fraction numerator 2 straight h over denominator sin squared straight theta end fraction equals fraction numerator 2 straight h over denominator open parentheses 4 straight h close parentheses squared end fraction open parentheses straight R squared plus 16 straight h squared close parentheses space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 2 straight h over denominator 16 straight h squared end fraction open parentheses straight R squared plus 16 straight h squared close parentheses space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator 1 over denominator 8 straight h end fraction open parentheses straight R squared plus 16 straight h squared close parentheses space space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator straight R squared over denominator 8 straight h end fraction plus fraction numerator 16 straight h squared over denominator 8 straight h end fraction space space space space space space space space space space space space space space space space space space space space space space equals fraction numerator straight R squared over denominator 8 straight h end fraction plus 2 straight h space space space space space space space space space space space space space space space space space space space space space space space space space space space end style

Answered by Faiza Lambe | 7th Aug, 2015, 11:46: AM

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