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The image contains a set of physics or thermodynamics-related calculations involving two velocities:

1. **Most probable velocity** (\(v_{mp}\))
2. **Velocity that maximizes energy** (\(v_{maxE}\))

Both of these velocities are calculated for two cities: Bogotá and Barranquilla, using specific formulas.

### Formulas used:
1. **Most probable velocity**:
   \[
   v_{mp} = c \left(1 - \frac{1}{k}\right)^{\frac{1}{2}}
   \]
2. **Velocity that maximizes energy**:
   \[
   v_{maxE} = c \left(\frac{k + 2}{k}\right)^{\frac{1}{2}}
   \]

### Given values:
- For **Bogotá**:
  - \( k = 1.88 \)
  - \( c = 2.19 \)
- For **Barranquilla**:
  - \( k = 2.75 \)
  - \( c = 4.78 \)

### Calculation steps:

#### Bogotá:
1. **\(v_{mp}\) for Bogotá**:
   \[
   v_{mp} = 2.19 \times \left(1 - \frac{1}{1.88}\right)^{\frac{1}{2}}
   \]
   
2. **\(v_{maxE}\) for Bogotá**:
   \[
   v_{maxE} = 2.19 \times \left(\frac{1.88 + 2}{1.88}\right)^{\frac{1}{2}}
   \]

#### Barranquilla:
1. **\(v_{mp}\) for Barranquilla**:
   \[
   v_{mp} = 4.78 \times \left(1 - \frac{1}{2.75}\right)^{\frac{1}{2}}
   \]
   
2. **\(v_{maxE}\) for Barranquilla**:
   \[
   v_{maxE} = 4.78 \times \left(\frac{2.75 + 2}{2.75}\right)^{\frac{1}{2}}
   \]

The process shows the application of these formulas for both cities, leading to different velocities based on the respective constants \( k \) and \( c \) for each city.

Would you like me to walk through the steps in detail or calculate the final results? 

Here are 5 related questions you might find useful:
1. How does changing the constant \(k\) affect the velocities?
2. What is the significance of the constants \(k\) and \(c\) in the context of these calculations?
3. How are the results for Bogotá and Barranquilla different, and why?
4. Can this approach be generalized for any value of \(k\) and \(c\)?
5. What would happen if \(k = 1\)? How would the formula simplify?

**Tip:** Understanding how constants impact a formula helps you see broader relationships in physics and thermodynamics.

Calculate the most probable velocity and the velocity that maximizes energy for Bogotá and Barranquilla using given values for k and c.

This page provides detailed calculations for finding the most probable velocity (v_mp) and the velocity that maximizes energy (v_maxE) using the given values for k and c for the cities of Bogotá and Barranquilla. Key physics concepts such as the Maxwell-Boltzmann distribution are applied.

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