Baseball Problem: Does a Ball Hit the Roof? Answer Here!

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SUMMARY

The discussion centers on a calculus problem involving a baseball thrown vertically from a height of 6 feet at an initial velocity of 75 miles per hour (110 ft/sec) in a domed stadium with a roof at 200 feet. To determine if the ball hits the roof, one must calculate the maximum height using the equations of motion under constant acceleration due to gravity (32 ft/sec²). The key steps involve deriving the height function and applying integration techniques to find the peak height of the ball.

PREREQUISITES
  • Understanding of calculus concepts, specifically integration and derivatives.
  • Familiarity with kinematic equations for vertical motion.
  • Knowledge of gravitational acceleration (32 ft/sec²).
  • Ability to manipulate and solve algebraic expressions.
NEXT STEPS
  • Learn how to derive the height function for projectile motion using calculus.
  • Study the integration of acceleration to find velocity and position functions.
  • Explore kinematic equations for vertical motion in physics.
  • Practice solving similar problems involving maximum height calculations in projectile motion.
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Students studying calculus, physics enthusiasts, and anyone preparing for calculus tests or seeking to understand projectile motion in a real-world context.

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Homework Statement


This is part of a calculus test and I am clearly missing something because this should be an easy problem: A baseball pitcher is having fun with the fans. He can throw a ball vertically froma point 6 feet above the ground at 75 miles per hour (110 ft/sec). He is standing in a domed stadium with a roof 200 feet above the ground. Does the ball hit the roof? Justify your answer.

We are currently doing integrals and just finished derivatives if that's any help. Any help would be appreciated, thanks.


The Attempt at a Solution


I sadly don't have one.
 
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Clearly you need to figure out the maximum height the ball reaches. Can you write an expression for the height as a function of time?

You can start with the acceleration due to gravity and integrate if you like.
 

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