question_answer 62)

                  A gun can fire shells with maximum speed \[{{\upsilon }_{0}}\] and die maximum horizontal range that can be achieved is \[R=\,\frac{\upsilon _{0}^{2}}{g}.\] If a target farther away by distance \[\Delta x\] (beyond R) has to be hit with the same gun (Fig.), show that it could be achieved by raising die gun to a height at least \[h=\,\Delta x\,\left[ 1+\,\frac{\Delta x}{R} \right]\]

Answer:

                  The motion of projectile thrown with velocity \[{{\upsilon }_{0}}\] at an angle \[\theta \] is given by                 \[y=\,x\,\tan \theta +\,\frac{g{{x}^{2}}}{2\upsilon _{0}^{2}\,{{\cos }^{2}}\theta }\]                        ?.. (i)                 \[R=\frac{\upsilon _{0}^{2}}{g}\,(\max .)\] if \[\theta =\,{{45}^{o}}\]                 \[y=x+\,\frac{g{{x}^{2}}}{\upsilon _{0}^{2}}\]                 \[y=x+\,\frac{{{x}^{2}}}{R}\]                       .. (ii)                 When \[y=h\] and \[x=R+\Delta x\] \[\therefore \] \[h=\,\Delta x\,\left[ 1+\,\frac{\Delta x}{R} \right]\].