Angle at which distance is longest?

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SUMMARY

The optimal angle for projectile motion to achieve maximum distance is 45 degrees when the launch and landing heights are equal. However, when the landing height differs, such as when a ball lands on a tower of height h, the angle must be recalculated. The formula to determine this angle involves the initial velocity v and the height difference between the launch and landing positions. For scenarios involving multiple towers, the angle must be adjusted to maximize the range to the farthest tower.

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  • Understanding of basic projectile motion principles
  • Familiarity with kinematic equations
  • Knowledge of trigonometric functions
  • Ability to manipulate algebraic formulas
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  • Study the kinematic equations for projectile motion with varying heights
  • Learn how to derive the angle of projection for different landing heights
  • Explore optimization techniques for maximizing range in projectile motion
  • Investigate the effects of air resistance on projectile trajectories
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Physics students, engineers, and anyone interested in understanding projectile motion dynamics, particularly in scenarios involving varying launch and landing heights.

Irrelativity
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I have a question about projectory motion.

I know that if ball is thrown by a person with a initial velocity v(only x and ycomponents), a maximum distance d (distance of x component) can be achived at angle of 45 degree (an angle between the initial velocity and x component of the velocity). But this is only true when the initial position of y component is the same as the terminal position of y conpoment. Here's the my question: how do you guys find the angle at which when the height of a landing position is not the same as the height of the initial position (i.e. Initial y is not equal to terminal y)? Let's say the ball lands on a tower which has a height h with minimum velocity. What's the formula to find the angle?
 
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You mean, what is the angle for the ball to land on the tower? Or to reach maximum range?
Or you have a series of towers and you want to reach the farthest tower possible?
 
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