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

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Discussion Overview

The discussion revolves around calculating the maximum height of a projectile using calculus, specifically focusing on two scenarios: one on Earth and another on the Moon. Participants explore the application of derivatives to find the time at which the projectile reaches its peak height based on given height functions.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant presents a height function for a paper clip launched on Earth, H(t) = 64t - 16t², and asks how to determine the time to reach maximum height.
  • Another participant suggests that the time to reach maximum height can be found by setting the derivative H'(t) to zero, leading to the equation 64 - 32t = 0, resulting in t = 2 seconds.
  • The second participant also notes that the initial velocity of the paper clip is 64 feet per second, clarifying the interpretation of the height function.
  • A third participant requests further clarification and examples of the solution process to understand how to arrive at the answer.
  • A fourth participant challenges the first poster's claim of having solved the problem, asking them to share their solution for verification.

Areas of Agreement / Disagreement

Participants express varying levels of understanding and willingness to share solutions, with some seeking clarification while others challenge the initial claims. No consensus is reached on the overall approach or solution process.

Contextual Notes

There is a lack of clarity regarding the initial conditions and the interpretation of the height functions. The discussion does not resolve how to calculate the maximum height on the Moon or the specifics of the second height function provided.

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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