How Do You Calculate Maximum Height in Projectile Motion Using Calculus?

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SUMMARY

The discussion focuses on calculating the maximum height of a projectile using calculus, specifically for a paper clip launched with a rubber band. The height function on Earth is given by H(t) = 64t - 16t², where the maximum height is reached at t = 2 seconds, resulting in H(2) = 64 feet. For the moon, the height function is H(t) = 64t - 2.6t², and participants are encouraged to derive the time to reach maximum height and the corresponding height using similar calculus techniques.

PREREQUISITES
  • Understanding of calculus, particularly derivatives
  • Familiarity with projectile motion equations
  • Knowledge of maximum and minimum values in functions
  • Ability to solve quadratic equations
NEXT STEPS
  • Learn how to derive maximum height using calculus for different projectile motion scenarios
  • Explore the differences in gravitational acceleration on various celestial bodies
  • Study the implications of initial velocity on projectile motion
  • Investigate the use of calculus in optimizing real-world projectile launches
USEFUL FOR

Students studying physics or calculus, educators teaching projectile motion, and anyone interested in applying calculus to real-world problems in mechanics.

orangesang
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Ok.
So I need two questions answered so i can check answers with what u guys got.


1. a. On Earth, you could easily shoot a paper clip 64 feet straight up into the air with a rubber band. After t seconds, the clip will be H(t)= 64t-16t^2 feet above you.

How long will it take the clip to reach its maximum height?

b. On the moon, an identical launch would send the clip to a height of H(t)=64t-2.6t^2

How long would it take for the clip to reach its maximum height and what would be that height?


Ok. So if you could show me how you did it also it would really help.
 
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The t value of the maximum will be a zero of H'(t) since that is when the paper clip stops going up and starts to go back down. Since H'(t) = 64-32t, 64-32t=0 => 64=32t => t=2 with H(2) = 64.

Also, you would be shooting the paperclip up at 64 feet per second from a height of 0 feet with that equation, not necessarily up 64 feet in the air.
 
orangesang said:
Ok. So if you could show me how you did it also it would really help.

Can you show us how you did it?
 
You initially said that you had already solved the problem and just wanted to check your answers.
Now you say you do not know how to do the problem.

How about showing YOUR solution so we can check it?
 

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