Why does a projectile have a maximum horizontal range when the angle of projection is 45 degrees?

Answer 1

Because at that angle, you have the best possible balance between the x-axis and y-axis giving the projectile maximum range.

Shooting the projectile at an angle higher than 45 degrees, will simply give more force to the y-axis, taking away from the x-axis/range.

Shooting it at an angle lower than 45 degrees, will not give the projectile enough height, and gravity will quickly pull the projectile down to earth and it will not go as far.

So 45 degrees strikes the best possible balance.

Answer 2

Why is always a tough question. The problem is a little more complex than can be answered with a single number. Factors which are normally ignored sometimes can make big differences. If you throw a projectile against a big wind then a less steep angle will result in the projectile going farther.
The simple situation where air can be ignored and the distances and terain are such that the earth can be considered flat then the 45 degree angle does result in the maximum horizontal distance. There are two important factors that are changing when the angle changes.
The height to which the projectile reaches and therefore the time the projectile is in the air is determined by the component of velocity that is perpendicular to the earth. The steeper the angle the greater percentage of the total velocity is vertical and the longer the projectile stays in the air.
The speed across the ground is deminished as the angle is increased.

The distance traveled is equivalent to speed multiplied by time.

If one threw a projectile straight up there would be zero horizontal velocity and so no mater how long the projectile stayed up it would fall back to the place that it started.
If one threw a projectile from the ground horizontaly it would not stay up an therefore it would not go any distance either.

45 degrees is half way between horizontal and vertical.

There is a branch of mathmatics called triginometry which explains the relationships between angles and the sides of a triangle. There is another branch of mathematics called calculus which explains rate of change kinds of problems.

Answer 3

Not quite so when air friction is factored in.
But when the air effect is made negligible you can argue by symmetry (because of the separability of longitudinal motion and vertical motion in the gravitational field) : 90 degrees gets the highest range, 0 degrees the longest (but gravitation is in the way); therefore the mid-point -- 45 degrees gets you the 'best of both' cases. In other words it would go forever if gravity was absent. That is the time is set by the climb up (inertia) and the fall down. At that time the object is traveling sideways by inertia. Giving it the largest inertia depends on an initial sideways velocity, which in turn is a cosine projection of the shooting angle.

Answer 4

Every projectile that encounters opposing force (such as gravity or friction) is going to have a maximum horizontal range despite the angle of launch.