Adam stretches a spring by some length. John stretches the same spring later by three times the length stretched by Adam. Find the ratio of the stored energy in the first stretch to that in the second stretch.

Answer

To solve the problem, we need to calculate the stored energy in the spring when stretched by Adam and when stretched by John, and then find the ratio of these two energies.

### Step-by-Step Solution:

1. **Define the Variables**:
- Let the length Adam stretches the spring be \( x \).
- Therefore, John stretches the spring by \( 3x \).

2. **Energy Stored in the Spring**:
- The formula for the potential energy stored in a spring when it is stretched or compressed is given by:
\[ U = \frac{1}{2} k x^2 \]
where \( U \) is the stored energy, \( k \) is the spring constant, and \( x \) is the amount of stretch or compression.

3. **Calculate Energy for Adam**:
- For Adam, who stretches the spring by \( x \):
\[ U_1 = \frac{1}{2} k x^2 \]

4. **Calculate Energy for John**:
- For John, who stretches the spring by \( 3x \):
\[ U_2 = \frac{1}{2} k (3x)^2 = \frac{1}{2} k \cdot 9x^2 = \frac{9}{2} k x^2 \]

5. **Find the Ratio of Energies**:
- Now, we need to find the ratio of the energy stored by Adam to that stored by John:
\[ \frac{U_1}{U_2} = \frac{\frac{1}{2} k x^2}{\frac{9}{2} k x^2} \]
- Simplifying this:
\[ \frac{U_1}{U_2} = \frac{1}{2} \cdot \frac{2}{9} = \frac{1}{9} \]

6. **Conclusion**:
- The ratio of the stored energy in the first stretch (by Adam) to that in the second stretch (by John) is:
\[ \frac{U_1}{U_2} = 1 : 9 \]

### Final Answer:
The ratio of the stored energy in the first stretch to that in the second stretch is \( 1 : 9 \).