Need help writing a joke equation

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SUMMARY

This discussion focuses on the physics of paintball markers and their effective shooting range, specifically addressing the misconception that more expensive markers shoot farther. The standard shooting velocity is set at 300 fps (approximately 100 m/s), and the calculated range from a height of 1.5 meters is about 55 meters. The user seeks to incorporate the price of the marker into the range equation, proposing a model where prices below 600 reduce the range and prices above 600 increase it, with specific limits on the adjustments. The suggestion includes using an arctangent function to achieve the desired scaling.

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  • Understanding of basic physics principles, particularly projectile motion
  • Familiarity with paintball equipment and terminology
  • Knowledge of mathematical functions, specifically arctangent
  • Ability to perform calculations involving velocity and distance
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  • Research projectile motion equations and their applications in sports
  • Learn about the arctangent function and its properties in mathematical modeling
  • Explore the impact of different variables on projectile range in physics
  • Investigate how price factors can influence performance metrics in competitive sports
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Paintball players, sports physicists, mathematicians interested in modeling, and anyone involved in the design or marketing of paintball equipment.

1MileCrash
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This is all done really fast for fun, may be some mistakes.

I'm a paintball player, and there seems to be a very prevalent misconception that certain paintball markers shoot paintballs greater distances than others.

This isn't true because before actually playing, your marker must be chronographed to be shooting 300 fps. Obviously if these two paintballs are roughly spherical, and are both traveling at similar velocities, physics doesn't care how expensive the paintball marker is.

Changing 300 fps to 100 m/s for simplicities sake, and assuming the paintball is shot from a height of 1.5 meters:

(I'm getting really frustrated with the math functions, so I'm typing it)

( 100 m/s ) √ (3 m / 9.8 m/s2)

Shows the distance, which comes out to about 55 meters. That seems close enough.

But I want to incorporate a new factor into the equation, the price of the marker.

We will say that anything below a value P of 600 will reduce the range given above, and anything above 600 will increase the range given above.

However at P = 100 the value should stop decreasing, and should decrease by no more than 30 meters. After P = 1600 the value should stop increasing, and should increase by no more than 50 meters.

Can anyone think of a simplistic and clean way to implement this? Perhaps modifying the 100 m/s?
 
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How about something like an arctangent?

Just start with 600 + arctan(x) and scale such that it gives you the properties you want.
 

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