Projectile Range: Why 45 Degrees?

[skull]

Does anyone know why, mathematically, a projectile gets the maximum range when launched at 45 degrees?

 

[Doc Al]

Does anyone know why, mathematically, a projectile gets the maximum range when launched at 45 degrees?

Can you derive the range of a projectile in terms of launch angle? (If not, read this: Range of Trajectory) Once you do that, you can find the maximum using basic calculus.

 

[m2003]

Yes, there is a mathematical explanation for why a projectile achieves maximum range when launched at 45 degrees. This phenomenon can be explained by considering the horizontal and vertical components of the projectile's velocity.

When a projectile is launched at an angle of 45 degrees, the initial velocity can be broken down into two components: a horizontal component and a vertical component. The horizontal component remains constant throughout the trajectory of the projectile, while the vertical component changes due to the effect of gravity.

At the peak of the projectile's trajectory, the vertical component of the velocity becomes zero, meaning that the projectile is momentarily moving only horizontally. This results in the projectile covering the maximum horizontal distance before it starts to descend due to the effect of gravity.

At any other angle, the vertical component of the velocity would not be zero at the peak of the trajectory, resulting in a shorter horizontal distance covered by the projectile.

Furthermore, at angles lower than 45 degrees, the horizontal component of the velocity decreases, resulting in a shorter horizontal distance covered. At angles higher than 45 degrees, the vertical component of the velocity increases, causing the projectile to descend more rapidly and covering a shorter horizontal distance.

Therefore, mathematically, launching a projectile at 45 degrees results in the maximum range due to the balance between the horizontal and vertical components of the velocity.