If the angles of projection of a projectile with same initial velocity exceed or fall short of 45∘ by equal amounts, then the ratio of horizontal ranges is1:21:31:41:1

The statement given in the question is a well-known fact in Physics, specifically in the topic of Projectile Motion.

The fact states that "If the angles of projection of a projectile with the same initial velocity exceed or fall short of 45 degrees by equal amounts, then the ratio of their horizontal ranges is 1:1."

This means that if one angle is (45 - x) degrees and the other is (45 + x) degrees, where x is the same for both angles, then the horizontal ranges (the horizontal distance covered by the projectile) for these two angles of projection will be the same.

This is because the range of a projectile launched at an angle θ with initial velocity u is given by R = (u²sin2θ)/g, where g is the acceleration due to gravity. The function sin2θ is symmetrical about θ = 45 degrees, which means that sin2(45 + x) = sin2(45 - x). Therefore, the ranges for angles (45 - x) and (45 + x) are equal, giving the ratio 1:1.