A small block of mass m is projected horizontally from the top of the smooth and fixed hemisphere of radius r = 8 m with speed u as shown. For values of u≥u0,(u0=gr) it does not slide on the hemisphere. [i.e., leaves the surface at the top itself] For u=2u0, it lands at point P on ground. Find OP:

mg=mu02r⇒u0=grNow, along vertical; r =12gt2⇒t=2rgAlong horizontal ; OP = 2u0t=22r

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A small block of mass m is projected horizontally from the top of the smooth and fixed hemisphere of radius r = 8 m with speed u as shown. For values of $u \geq u_{0}, (u_{0} = \sqrt{g r})$ it does not slide on the hemisphere. [i.e., leaves the surface at the top itself] For $u = 2 u_{0},$ it lands at point P on ground. Find OP:

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Detailed Solution

$m g = \frac{m u_{0}^{2}}{r} \Rightarrow u_{0} = \sqrt{g r}$

$Now, along vertical; r = \frac{1}{2} g t^{2} \Rightarrow t = \sqrt{\frac{2 r}{g}}$

${Along horizontal ; OP = 2u}_{0} t = 2 \sqrt{2} r$

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