\begin{tabular}{l} During a family picnic, Harper ran across a flat field to the base of an ash tree. \ She reached up and pulled herself onto the lowest branch of the tree. Then, she \ climbed to a higher branch and stopped to look around. From the branch, she \ could see her stepdad waving to her. \ When was the most gravitational potential energy stored between Harper and Earth? Assume \ that Harper's mass did not change. \ when Harper was running toward the tree \ when Harper first reached up to grab a branch \ when Harper saw her stepdad wave to her \ wher \ wher \ \hline\end{tabular}

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To determine when Harper had the **most gravitational potential energy** relative to the Earth, let's analyze each stage of her actions:

1. **Running toward the tree:**
   - **Height (`h`)**: Lowest point (ground level).
   - **Gravitational Potential Energy (`U = mgh`)**: Minimal, since `h` is smallest.

2. **First reached up to grab a branch:**
   - **Height (`h`)**: Increased from ground level.
   - **Gravitational Potential Energy (`U = mgh`)**: Higher than when running, but not the highest.

3. **Climbed to a higher branch and saw her stepdad:**
   - **Height (`h`)**: Highest point among the three scenarios.
   - **Gravitational Potential Energy (`U = mgh`)**: Maximum, as `h` is largest at this stage.

**Conclusion:**
Gravitational potential energy increases with height. Therefore, Harper had the most gravitational potential energy **when she climbed to the higher branch and saw her stepdad waving to her**.

**Answer:** When Harper saw her stepdad wave to her from the higher branch.
Harper had the most gravitational potential energy when she climbed to the higher branch and saw her stepdad wave to her.

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