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To calculate the initial velocity \( v_0 \) for the marble launched vertically, we can use the concept of hangtime and kinematic equations.

### Steps to Solve

1. **Record Hangtime**: Perform three trials, recording the time the marble spends in the air (the "hangtime") for each trial in seconds.

2. **Calculate Average Hangtime**: Take the average of the three recorded times.

3. **Using the Hangtime Equation**:
   Since the marble is launched vertically and returns to its original height, the total hangtime (\( t \)) includes both the ascent and descent. The time to reach the peak is \( t_{up} = t/2 \), and we can use the kinematic equation for velocity:
   
   \[
   v_0 = g \cdot t_{up}
   \]
   
   where:
   - \( v_0 \) is the initial velocity,
   - \( g \) is the acceleration due to gravity (approximately \( 9.81 \, \text{m/s}^2 \)),
   - \( t_{up} \) is half of the hangtime (i.e., \( t_{up} = \text{average hangtime} / 2 \)).

4. **Calculate Initial Velocity**:
   After computing \( t_{up} \), plug it and \( g \) into the equation to find \( v_0 \).

### Equation Summary
The initial velocity \( v_0 \) is:
\[
v_0 = \frac{g \cdot (\text{average hangtime})}{2}
\]

Would you like assistance in filling out the table with values, or do you have specific questions about the calculations?

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### Further Related Questions

1. What are the potential sources of error in measuring hangtime, and how can they affect the calculation of initial velocity?
2. How would the initial velocity change if the launcher angle was adjusted away from vertical?
3. How does air resistance affect the motion of the marble, and is it significant in this experiment?
4. Can you explain how the value of \( g \) was chosen and why it might vary slightly depending on location?
5. What would be the effect on the hangtime if the launch velocity was increased?

### Tip:
Always record the exact time measurements carefully to reduce errors in calculating average hangtime and initial velocity.

Part 1: Finding initial velocity. Set your launcher on the floor and aim the launcher straight up into the air. Record the time when the marble is launched and returns to its initial height. Use the average hangtime to calculate the initial velocity.

Learn how to calculate the initial velocity of a vertically launched marble by measuring its hangtime. This physics problem involves kinematic equations, including the formula v_0 = g * (average hangtime / 2), and is suitable for students in Grades 10-12.

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