Question

A particle is projected with velocity 2√(gh) ,so that it just clears two wall  of equal height h which are at a distance of 2h from each other.show that the time of passing between the walls is  2√(h/g)......

Answer 1

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See diagram.

[tex] h = 2\sqrt{gh}\ \ Sin\ \theta\ t\ -\ \frac{1}{2}\ g\ t^2 \\ \\ t^2 - 4 \sqrt{\frac{h}{g}}\ Sin\ \theta\ t + \frac{2h}{g} = 0 \\ \\ [/tex]

[tex]Delta = \frac{16h}{g}\ Sin^2\ \theta - \frac{8h}{g} = \frac{8h}{g} [2 Sin^2\ \theta - 1] \\ \\ Delta = \frac{8h}{g} Cos\ (180-2\theta) \\ \\ t2\ -\ t1\ =\ difference\ of\ roots\ = \sqrt{Delta} = \sqrt{\frac{8h\ Cos\ (180-2\theta)}{g}} \\ \\ [/tex]

The projectile traveled 2h distance in t2 - t1  horizontally. see diagram.

[tex] 2\sqrt{gh}\ Cos\ \theta * \sqrt{\frac{8h\ Cos\ (180-2\theta)}{g}} = 2h \\ \\ Cos\ \theta \sqrt{8 Cos(180-2\theta)} = 1 \\ \\ 8\ cos (180-2\theta) = Sec^2\theta \\ 8 [- cos2\theta ] = 1 + tan^2\theta \\ \\ -8 \frac{1-tan^2\theta}{1+tan^2\theta} = 1+tan^2\theta \\ \\ tan^4\theta -6tan^2\theta +9 = 0 \\ \\ tan^2\theta = 3 \\ \\ tan\ theta = \sqrt{3} \\ \theta = 60deg, \\ \\ Substituting\ in\ t2-t1 = \sqrt{\frac{8h}{g}cos\ 60} = 2\sqrt{\frac{h}{g}}\\ \\ [/tex]

Answer 2

Please see simple answer from Rahul Sardana's Book