How do momentum, impulse, and GPE factor into solving physics questions?

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Momentum, impulse, and gravitational potential energy (GPE) are crucial concepts in solving physics problems, especially in competitive contexts. Understanding the relationships between these concepts can aid in analyzing motion and energy transfer in various scenarios. Participants in the discussion emphasize the importance of breaking down problems step by step to identify where confusion arises. Clarifying the application of formulas related to momentum and GPE can significantly enhance problem-solving skills. Seeking assistance and verifying calculations is essential for mastering these physics concepts.
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Hey guys I am preparing for a physics contest and I am stuck with on a few questions. they are step by step questions. If someone can clearly explain how how to solve the following questions it would greatly help. Thank you.
 

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Please show what you've done so far and point out where you are stuck.
 
this is what I've got so far but I am not sure if its right.

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thats what I've figured out so far but i need to verify this.

please help.
 
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