Section 5.8: Explosions and Disintegration

Explosions and disintegration involve a system of particles separating rapidly. The key is that momentum of the system is conserved if external forces are negligible.

Conservation of Momentum:
  • Total momentum before = Total momentum after
  • \( \sum \vec{p}_{before} = \sum \vec{p}_{after} \)
  • Internal forces (explosion forces) cancel in the total momentum sum.

Example 1

A 10 kg particle at rest explodes into two fragments of masses 4 kg and 6 kg. If the 4 kg fragment moves at 3 m/s to the right, find the velocity of the 6 kg fragment.

Total momentum before: 0

Let \(v_2\) be velocity of 6 kg fragment:

\(0 = 4*3 + 6*v_2 \Rightarrow v_2 = -2 \text{ m/s (left)}\)

Example 2

Two particles of 2 kg and 3 kg moving with velocities 4 m/s and -2 m/s collide and stick together. Find final velocity of combined mass.

\( v_f = \frac{2*4 + 3*(-2)}{2+3} = \frac{8-6}{5} = 0.4 \text{ m/s} \)

Practice Problems

  1. A 12 kg object at rest explodes into two pieces of 5 kg and 7 kg. The 5 kg piece moves at 4 m/s. Find velocity of 7 kg piece.
  2. 3 kg and 2 kg particles explode along x-axis. Initial velocity 0. Final velocities: 3 kg fragment = 5 m/s right. Find 2 kg fragment velocity.
  3. Two objects collide inelastically: m1 = 4 kg, v1=3 m/s; m2=6 kg, v2=-2 m/s. Find final velocity.
  4. Firework explodes into 3 equal masses along x-axis. First moves 6 m/s, second 4 m/s opposite. Find third mass velocity.
  5. Two ice skaters push off each other. m1=50 kg, v1=1.2 m/s; m2=unknown mass. v2=0.8 m/s. Find mass of second skater.
  6. A cannon fires projectile (recoil considered). Find recoil velocity of cannon if projectile mass & velocity known.
  7. Particle system initially at rest. Explodes in x-y plane: compute fragment velocities given angles & masses.
  8. Explosion of stationary object into unequal masses in 2D. Solve for unknown velocity using momentum conservation.
  9. Two masses connected by spring explode along horizontal. Find velocities post-explosion.
  10. Rocket burns fuel in space. Neglect external forces. Show momentum conservation for rocket and expelled fuel.
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