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To solve this problem, we analyze the conservation of momentum. A stationary object means its initial momentum is zero. Upon explosion, the total momentum of the system remains zero, so the vector sum of the momenta of all fragments must also be zero. Here's the step-by-step solution:

### Problem Setup
- **Masses**: \( 3m \), \( 4m \), and \( 5m \).
- The two pieces move with momenta \( 3mv \) upward and \( 4mv \) to the right, as shown in the diagram.
- The third piece (\( 5m \)) has momentum \( p_3 \) that must cancel out the total momentum of the other two pieces.

### Step 1: Total Momentum of the First Two Pieces
Momentum is a vector quantity, so we sum the momenta in the \( x \)- and \( y \)-directions separately.

- **In the \( x \)-direction**: Only the second piece contributes with \( p_x = 4mv \).
- **In the \( y \)-direction**: Only the first piece contributes with \( p_y = 3mv \).

### Step 2: Momentum of the Third Piece
The momentum of the third piece (\( p_3 \)) must counteract the combined momentum of the other two pieces. Using the conservation of momentum:
\[
\vec{p}_1 + \vec{p}_2 + \vec{p}_3 = 0
\]
This implies:
\[
\vec{p}_3 = -(\vec{p}_1 + \vec{p}_2)
\]
Thus:
- **In the \( x \)-direction**: \( p_{3x} = -4mv \).
- **In the \( y \)-direction**: \( p_{3y} = -3mv \).

### Step 3: Magnitude of Momentum of the Third Piece
The magnitude of \( \vec{p}_3 \) is:
\[
p_3 = \sqrt{p_{3x}^2 + p_{3y}^2}
\]
Substitute:
\[
p_3 = \sqrt{(-4mv)^2 + (-3mv)^2} = \sqrt{16m^2v^2 + 9m^2v^2} = \sqrt{25m^2v^2} = 5mv
\]

### Step 4: Velocity of the Third Piece
The mass of the third piece is \( 5m \), so its velocity is:
\[
v_3 = \frac{p_3}{m_3} = \frac{5mv}{5m} = v
\]

Thus, the velocity of the third piece is \( v \).

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If you'd like further clarification or details on any step, let me know! Here are related questions to expand your understanding:

1. What happens if the initial object is moving instead of stationary?
2. How does the angle between the momenta of the two fragments affect the result?
3. How would the problem change if the masses of the pieces were not integers?
4. Can momentum conservation be used to solve rotational motion problems?
5. How is energy conservation different from momentum conservation in explosions?

**Tip**: Always analyze momentum as a vector quantity, breaking it into components if necessary. This ensures you account for all directions correctly!

A stationary object explodes, breaking into three pieces of masses 3m, 4m, and 5m. The two pieces of mass 3m and 4m move off as shown by their momenta. What is the velocity of the third object with mass 5m?

Explore a detailed solution to a physics problem involving momentum conservation. A stationary object explodes into three fragments of masses 3m, 4m, and 5m. Learn how to calculate the velocity of the third fragment using vector addition and kinematics.

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