https://r2.erweima.ai/mathbot/image/1719322765213olss5beu.jpg

https://r2.erweima.ai/mathbot/image/1719322765215fraxpypy.jpg

I see two images uploaded, one of which is of a cat, and the other appears to be a page from a physics book or worksheet involving relativistic equations. Let's focus on the physics problem you have shared. Here's an analysis of the mathematical content visible in the image.

### Analyzing the Problem

The equation provided appears to relate to the Lorentz factor, \(\gamma\), which is used in special relativity. The Lorentz factor is given by:

\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

Where:
- \(v\) is the velocity of the moving object.
- \(c\) is the speed of light.

In the image, we can see the following equations:

1. \[ \gamma = \frac{1}{\sqrt{1 - \frac{0.81c^2}{c^2}}} = \frac{1}{\sqrt{1 - 0.81}} \]
2. \[ \gamma = \frac{1}{\sqrt{0.19}} \]
3. \[ l = l_0 \sqrt{0.19} \Rightarrow l = l_0 \]

It seems like the problem involves calculating the contracted length \(l\) of an object moving at a certain velocity.

### Length Contraction

The length contraction formula is given by:

\[ l = \frac{l_0}{\gamma} \]

where:
- \(l_0\) is the proper length (length of the object in the rest frame).

Let's go through the steps indicated in the circled part:

### Step-by-Step Solution

1. **Finding \(\gamma\):**

   \[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]

2. **Substitute given values:**

   In the image, \( \gamma = \frac{1}{\sqrt{0.19}} \).

3. **Length Contraction:**

   Given:
   \[ l = l_0 \sqrt{0.19} \]

   Rearrange to solve for \(l_0\):
   \[ l = \frac{l_0}{\gamma} \]

   Substitute \(\gamma\) into the equation to find \(l\).

### Conclusion

If you have any specific question or step in the problem that you'd like to focus on, please let me know! Here are some possible questions you might have:

1. What is the Lorentz factor and how is it derived?
2. How does length contraction work in special relativity?
3. Can you explain the significance of \(\gamma\) in relativistic equations?
4. How do you calculate the proper length \(l_0\) from the contracted length \(l\)?
5. What are some real-world applications of length contraction and the Lorentz factor?

**Tip:** Always ensure to substitute the correct values into the equations and simplify step by step to avoid calculation errors.

Explore the Lorentz factor and length contraction in special relativity with a detailed analysis. This problem involves calculating the Lorentz factor and understanding its implications on length contraction. Suitable for advanced high school or college students.

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