Projectile Range: Why 45 Degrees?

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A projectile achieves maximum range when launched at a 45-degree angle due to the optimal balance between vertical and horizontal velocity components. The range can be derived mathematically by analyzing the projectile's trajectory and applying basic calculus to find the maximum point. At 45 degrees, the sine and cosine functions are equal, maximizing the product of the launch velocity components. This principle is fundamental in physics and can be explored further in resources like the "Range of Trajectory." Understanding these concepts is essential for optimizing projectile motion in various applications.
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Does anyone know why, mathematically, a projectile gets the maximum range when launched at 45 degrees?
 
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skull said:
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.
 

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