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To solve this problem, we can apply the conservation of momentum. Since the explosion occurs on a frictionless surface, there are no external forces acting on the system, so the total momentum before and after the explosion must be equal.

### Given:
- Total mass \( m = 2.6 \, \text{kg} \)
- After the explosion, the mass splits into two parts:
  - Mass 1: \( m_1 = 1.3 \, \text{kg} \), moving at \( v_1 = 3.1 \, \text{m/s} \) (due north)
  - Mass 2: \( m_2 = 1.3 \, \text{kg} \), moving at \( v_2 = 5.7 \, \text{m/s} \) at an angle of \( 27^\circ \) north of east.

### Solution:

1. **Calculate the momentum of each fragment after the explosion in vector form.**
   - For **Mass 1** (north):
     - \( p_{1y} = m_1 \cdot v_1 = 1.3 \times 3.1 = 4.03 \, \text{kg} \cdot \text{m/s} \) (in the positive y-direction)
     - \( p_{1x} = 0 \) (since it moves purely north)

   - For **Mass 2** (27° north of east):
     - \( p_{2x} = m_2 \cdot v_2 \cdot \cos(27^\circ) = 1.3 \times 5.7 \times \cos(27^\circ) \)
     - \( p_{2y} = m_2 \cdot v_2 \cdot \sin(27^\circ) = 1.3 \times 5.7 \times \sin(27^\circ) \)

2. **Total initial momentum (before explosion):**
   - Let \( v \) be the initial speed of the 2.6 kg kit.
   - Total initial momentum \( P_{\text{initial}} = m \cdot v \).

3. **Set up equations for momentum conservation in each direction:**
   - In the **x-direction**: \( m \cdot v_x = p_{2x} \)
   - In the **y-direction**: \( m \cdot v_y = p_{1y} + p_{2y} \)

4. **Solve for \( v \) using \( v = \sqrt{v_x^2 + v_y^2} \).**

Let me compute these values for you.The original speed of the mess kit before it exploded was approximately \( 3.81 \, \text{m/s} \).

Would you like further details on the solution steps or have any questions?

Here are five related questions you might consider:
1. How would the result change if the surface were not frictionless?
2. What if the masses were unequal after the explosion?
3. How would we approach this if one part moved at a different angle?
4. What does conservation of momentum tell us about closed systems?
5. How does momentum conservation apply in two-dimensional collisions?

**Tip:** For problems involving angles, breaking vectors into components is essential for accuracy.

A 2.6 kg mess kit sliding on a frictionless surface explodes into two 1.3 kg parts, one moving at 3.1 m/s, due north, and the other at 5.7 m/s, 27° north of east. What is the original speed of the mess kit?

Explore how to find the original speed of a 2.6 kg mess kit before it exploded on a frictionless surface. Using conservation of momentum, we break down the movement into vector components and calculate the initial speed with detailed steps. This problem involves trigonometry and vector analysis, suitable for students in Grades 11-12.

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