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A figher plane flying horizontally at an...

A figher plane flying horizontally at an altitude of 1.5 km with speed `720km//h` passes directly overhead an anti-aircraft gun. At what angle from the vertical should the gun be fired for the shell with muzzle speed `600ms^(-1)` to hit the plane? At what minium altitue should the pilot fly the plane to avoid being hit? `[Take g=10`ms^(-2))`

Text Solution

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Let G be the position of the anti-aircraft gun and E that of the enemy plane flying horizontally with a speed (V) = 300 m/s when a shell is fired with a speed u = 600 m/s in a direction 0 with the horizontal (in figure). The horizontal component of u is u cos 0.
Let the shell hit the plane at point T and let t be the time taken for the shell to hit the plane. It is clear that the shell will hit the plane, if the horizontal distance ET traveled by the plane (moving with speed V) in time t is equal to the distance traveled in the horizontal direction by the shell (moving with speed u cos 0) in time t, i.e., if
`V xx t=u cos 0 xx t`
or `V= u cos 0`
`rarr cos 0 =(V)/(u) =(300 m//s)/(600 m//s) =(1)/(2) =0.5" or "0=60^(@)`
Therefore, angle with the vertical `=90^(@)-0 =30^(@)`
Thus, the shell should be fired in a direction `30^(@)` with the vertical.
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