Math Problem Statement

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?

Solution

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: 3m3m, 4m4m, and 5m5m.
  • The two pieces move with momenta 3mv3mv upward and 4mv4mv to the right, as shown in the diagram.
  • The third piece (5m5m) has momentum p3p_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 xx- and yy-directions separately.

  • In the xx-direction: Only the second piece contributes with px=4mvp_x = 4mv.
  • In the yy-direction: Only the first piece contributes with py=3mvp_y = 3mv.

Step 2: Momentum of the Third Piece

The momentum of the third piece (p3p_3) must counteract the combined momentum of the other two pieces. Using the conservation of momentum: p1+p2+p3=0\vec{p}_1 + \vec{p}_2 + \vec{p}_3 = 0 This implies: p3=(p1+p2)\vec{p}_3 = -(\vec{p}_1 + \vec{p}_2) Thus:

  • In the xx-direction: p3x=4mvp_{3x} = -4mv.
  • In the yy-direction: p3y=3mvp_{3y} = -3mv.

Step 3: Magnitude of Momentum of the Third Piece

The magnitude of p3\vec{p}_3 is: p3=p3x2+p3y2p_3 = \sqrt{p_{3x}^2 + p_{3y}^2} Substitute: p3=(4mv)2+(3mv)2=16m2v2+9m2v2=25m2v2=5mvp_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 5m5m, so its velocity is: v3=p3m3=5mv5m=vv_3 = \frac{p_3}{m_3} = \frac{5mv}{5m} = v

Thus, the velocity of the third piece is vv.


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!

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Math Problem Analysis

Mathematical Concepts

Momentum Conservation
Vector Addition
Kinematics

Formulas

Conservation of Momentum: p_initial = p_final
Magnitude of a Vector: |p| = sqrt(px^2 + py^2)
Velocity: v = p / m

Theorems

Conservation of Linear Momentum

Suitable Grade Level

Grades 10-12