At what angle does a missile have to be launched to cover the greatest distance.

The path taken by by object projected in any upward
direction other than the vertical, moves under the influence of two forces. These are
the momentum of the object due to the velocity at the time of launch of the object, and
the force of gravity. In this way the object takes a parabolic path moving upward as
well as forward in horizontal direction. The horizontal velocity of the object,
neglecting the air resistance remains constant, but due to the force of gravity the
upward velocity gets reduced till the object reaches a maximum height, when the velocity
becomes zero. Beyond this point the object continues to move forward in horizontal
direction at uniform velocity but with increasing downward velocity. This continues till
the object hit the ground.

The total horizontal distance
covered by the object for a given velocity of launch depends on two things. The
horizontal component of the velocity and the time for the object to hit the ground after
launch. This horizontal component of velocity is maximum in the horizontal direction,
and reduces as the angle of launch with respect to horizontal is increased. At 90
degrees the horizontal component of velocity reduces to zero. On the other hand the
height to which the object will rise and the total time for which it will remain
airborne is zero at horizontal angle of zero, and maximum at angle of 90
degrees.

The total horizontal distance travelled by the
object is thus maximum at a point in between the two. This point occurs at an angle of
launch of 45 degrees to the horizontal.