A fighter plane flying horizontally at an altitude of 1.5 km with speed 720 km/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 600 m s¯¹ to hit the plane ? At what minimum altitude should the pilot fly the plane to avoid being hit ? (Take g = 10 m s¯² ).
Given: the horizontal speed of the plane is 720 km / h , the height of the fighter plane is 1.5 km and the muzzle speed of the shell is 600 m / s .
Let θ be the angle with the vertical at which the gun is fired, t be the time taken by the shell to hit the plane, u x be the horizontal component of the velocity and u y be the vertical component of the velocity and v be the horizontal speed of the fighter plane.
The following figure shows the motion of the fighter plane.
The plane will be hit by the gun if the distance travelled by the plane and the shell is equal, i.e.,
u x t = v t u sin θ = v
Substitute the values in the above expression.
600 sin θ = 720 × 1000 60 × 60 sin θ = 1 3 θ = 19.47 °
The maximum height of the flight of the shell is given by,
h = u 2 cos 2 θ 2 g
Substitute the values in the above expression.
h = u 2 ( 1 − sin 2 θ ) 2 g = 600 2 ( 1 − 1 3 2 ) 2 ( 10 ) = 16000 m = 16 km
In order to avoid being hit by the shell of the gun, the pilot must fly the plane at minimum altitude, which is equal to the maximum height achieved by the shelli.e., at h = 16 km .
Given: the horizontal speed of the plane is
Let
The following figure shows the motion of the fighter plane.
The plane will be hit by the gun if the distance travelled by the plane and the shell is equal, i.e.,
Substitute the values in the above expression.
The maximum height of the flight of the shell is given by,
Substitute the values in the above expression.
In order to avoid being hit by the shell of the gun, the pilot must fly the plane at minimum altitude, which is equal to the maximum height achieved by the shelli.e., at
