Question

A projectile has the same range R when the maximum height attained by it is either H1 or H2. Find the relation between R, H1 and H2.

Answer 1

We know that :-

Range = R

$= > R = \frac{ {u}^{2}sin2 \alpha }{g}$
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For equal ranges body should be Projected with angle "alpha" and "90-alpha" from the Horizontal and for the angle :-

$= > H _{1} = \frac{ {u}^{2}. {sin}^{2} \alpha }{2g} \: \: \: AND \: \: \\ \\ = > H _{2} = \frac{ {u}^{2}. {cos}^{2} \alpha }{2g} \\ \\ THEREFORE \\ \\ = > H _{1}.H _{2} = \frac{ {u}^{4 } . {sin}^{2} \alpha . {cos}^{2} \alpha }{4 {g}^{2} } \\ \\ = H _{1}.H _{2} = \frac{1}{16} {( \frac{ {u}^{2}sin2 \alpha }{g}) }^{2} \\ \\ = > 16H _{1}.H _{2} = {R}^{2} \\ \\ = > R = 4 \sqrt{H _{1} .H _{2} }$
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Answer 2

U can always remember this relation that range=4× max height × cot( angle of projection) for easier calculation. Have a good day sis.