question_answer 63)

                  A particle is projected in air at an angle \[\beta \] to a surface which it self is inclined at an angle \[\alpha \] to the horizontal (Fig.). (a) Find all expression of range on the plane surface (distance oil the plane from the point of projection at which particle will hit the surface).                   (b) Time of flight.                 (c) \[\beta \] at which range will be maximum.

Answer:

                  Using, \[y=\,{{\upsilon }_{0y}}t+\frac{1}{2}\,{{a}_{y}}{{t}^{2}},\,\] we get                 \[0\,=({{\upsilon }_{0}}\,\sin \,\beta )\,t-\frac{1}{2}\,(g\,\cos \theta ){{t}^{2}}\]                 \[\therefore \,\,t=\,\frac{2{{\upsilon }_{0}}\sin \beta }{g\,\cos \,\alpha }\]                 Which is the time of flight.                 Now \[OQ\,={{\upsilon }_{0}}\,\cos \,(\alpha +\,\beta )\,\times t\]                 \[=\frac{{{\upsilon }_{0}}\cos (\alpha +\beta )\,\times \,2\,{{\upsilon }_{0}}\sin \,\beta }{g\,\cos \theta }\]                 \[\therefore \]\[R=\,OP\,\,=\,\frac{0Q}{\cos \alpha }\]                 \[=\,\frac{2\upsilon _{0}^{2}\,\sin \beta \,\,\cos \,(\alpha +\beta )}{g\,{{\cos }^{2}}\alpha }\]                 R will be maximum at \[\beta \] if                 \[\frac{dR}{d\beta }=\,0\] or \[\,\frac{2{{\upsilon }_{0}}}{g\,{{\cos }^{2}}\alpha }\,[\cos \,\beta \,\cos \,\,(\alpha +\,\beta )\]                 \[-\,\sin \,\beta \,\sin \,(\alpha +\,\beta )]=0\]                 or \[\tan \,(\alpha +\,\beta )\,=\,\cot \,\beta \,=\,\tan \,({{90}^{o}}-\beta )\]                 or \[\alpha +\,\beta \,=\,{{90}^{o}}\,-\beta \]                 or \[\beta ={{45}^{o}}\,-\frac{\alpha }{2}\,=\,\frac{\pi }{4}-\,\frac{\alpha }{2}.\]