Section 4.6: Explosions

An explosion occurs when an object breaks into two or more fragments due to an internal force. While the total momentum of the system is conserved, the kinetic energy usually increases, supplied by the release of stored energy (chemical, nuclear, elastic, etc.).

\[ m_1 v_{1f} + m_2 v_{2f} + \cdots = 0 \quad \text{(if initially at rest)} \]

If the system starts at rest, the vector sum of all fragment momenta must be zero. This ensures conservation of momentum.

Example 1

A shell of mass 10 kg explodes into two fragments: 6 kg and 4 kg. If the 6 kg piece moves at 3 m/s to the right, find the velocity of the 4 kg piece.

Conservation of momentum:
\( (6)(3) + (4)v = 0 \)
\( 18 + 4v = 0 \Rightarrow v = -4.5 \, \text{m/s} \)
The 4 kg fragment moves left at 4.5 m/s.

Example 2

A firework of 2 kg at rest explodes into three pieces: two of 0.5 kg each moving at 6 m/s and -6 m/s, and one piece of 1 kg. Find the velocity of the 1 kg piece.

Momentum before = 0.
Momentum after: \( (0.5)(6) + (0.5)(-6) + (1)(v) = 0 \)
\( 3 - 3 + v = 0 \Rightarrow v = 0 \).
The 1 kg piece remains at rest.

Practice Problems

  1. A rocket initially at rest explodes into two parts of equal mass. If one moves at 20 m/s, what is the velocity of the other?
  2. A 12 kg bomb breaks into two fragments: 8 kg moving at 2 m/s and 4 kg fragment moving in the opposite direction. Find its velocity.
  3. A firecracker explodes into three equal 0.2 kg fragments. Two fly off at 4 m/s east and 4 m/s north. Find the velocity (magnitude and direction) of the third fragment.
  4. A spacecraft ejects a 100 kg fuel tank backward at 50 m/s. If the spacecraft mass is 1000 kg, find its recoil velocity.
  5. A 5 kg object at rest explodes into two parts of 3 kg and 2 kg. If the 3 kg part moves at 10 m/s, find the velocity of the 2 kg part.
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