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Projectile Range and Optimal Launch Angles Explained
Chapter 3, Problem 15(choose chapter or problem)
For a fixed initial speed, the range of a projectile is determined by the angle at which it is fired. For all but the maximum, there are two angles that give the same range. Considering factors that might affect the ability of an archer to hit a target, such as wind, explain why the smaller angle (closer to the horizontal) is preferable. When would it be necessary for the archer to use the larger angle? Why does the punter in a football game use the higher trajectory?
Questions & Answers
QUESTION:
For a fixed initial speed, the range of a projectile is determined by the angle at which it is fired. For all but the maximum, there are two angles that give the same range. Considering factors that might affect the ability of an archer to hit a target, such as wind, explain why the smaller angle (closer to the horizontal) is preferable. When would it be necessary for the archer to use the larger angle? Why does the punter in a football game use the higher trajectory?
ANSWER:Step 1 of 2
We'll delve into the relationship between the launch angle and the distance covered by a projectile, taking an archer's arrow as an example. A projectile launched with initial velocity 'v' at an angle ‘\(\alpha\)', has a range '\(X_{\max }\)' given by this formula:
\(X_{\max }=\frac{V^{2} \sin 2 \alpha}{g}\)
In this equation, 'v' is the initial velocity, 'g' is the acceleration due to gravity, and '\(\alpha\)' is the launch angle
The maximum range is achieved when the launch angle '\(\alpha\)' is 45 degrees, meaning the \(\sin (2 \alpha)\) equals 1
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Projectile Range and Optimal Launch Angles Explained
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Join us in this enlightening video as we explore the fascinating world of projectiles and their optimal launch angles. We dive into the physics behind how a fixed initial speed affects the range of a projectile and why, for most angles, there are two choices that yield the same range. Delve into factors that affect an archer's ability to hit a target, including wind, and learn why a smaller angle, closer to the horizontal, is often preferable. Discover when and why a larger angle may be necessar