3.8 Momentum and Explosive Forces

Explosive forces are a dramatic example of the Law of Conservation of Momentum. An explosion is an event where a single object, initially at rest or in motion, breaks apart into multiple fragments due to strong internal forces. Even though the kinetic energy of the system increases dramatically, the total linear momentum of the system remains unchanged, provided there are no external forces acting on it.

1. The Principle of Momentum Conservation

For an isolated system (one with no net external forces), the total momentum before an event is equal to the total momentum after the event.

${\vec{p}}_{initial}={\vec{p}}_{final}$

In the case of an explosion, the forces involved are internal to the system (the object and its fragments). Therefore, the momentum of the entire system is conserved.

2. Explosion from a State of Rest

This is the simplest case to analyze. If an object is initially at rest, its total initial momentum is zero.

${\vec{p}}_{initial}=0$

After the explosion, the object breaks into multiple fragments, each with its own mass and velocity. According to the conservation of momentum, the vector sum of the momenta of all the fragments must still be zero.

${\vec{p}}_{final}=m_1{\vec{v}}_1+m_2{\vec{v}}_2+m_3{\vec{v}}_3+\cdots =0$

Example: A Bomb Exploding into Two Pieces

If a stationary bomb explodes into two fragments, their momenta must be equal in magnitude and opposite in direction.

• Initial momentum: ${\vec{p}}_i=0$
• Final momentum: $m_1{\vec{v}}_1+m_2{\vec{v}}_2=0$
• This leads to: $m_1{\vec{v}}_1=-m_2{\vec{v}}_2$

The lighter fragment will fly off with a much higher speed than the heavier fragment to ensure the momenta are balanced.

3. Recoil of a Rifle

The firing of a bullet from a rifle is a perfect example of this principle.

• System: The rifle and the bullet.
• Initial State: Before firing, both are at rest, so the total initial momentum is zero.

${\vec{p}}_i=0$

• Final State: After firing, the bullet of mass $m$ moves forward with velocity $\vec{v}$, and the rifle of mass $M$ recoils backward with velocity $\vec{V}$.

${\vec{p}}_f=m\vec{v}+M\vec{V}$

• Conservation of Momentum:

$0=m\vec{v}+M\vec{V}$

Solving for the recoil velocity of the rifle:

$\vec{V}=-\frac{m}{M}\vec{v}$

The negative sign indicates that the rifle's velocity is in the opposite direction to the bullet's velocity. Since the mass of the rifle ($M$) is much larger than the mass of the bullet ($m$), its recoil velocity ($\vec{V}$) is much smaller.

4. Explosion of a Moving Object

If an object is already moving when it explodes, its initial momentum is not zero. The total momentum of the fragments after the explosion must be equal to the momentum of the object just before it exploded.

Example: A Shell Exploding in Mid-Air

A shell with mass $M$ is moving with velocity ${\vec{V}}_{initial}$ when it explodes into two pieces, $m_1$ and $m_2$.

• Initial Momentum: ${\vec{p}}_i=M{\vec{V}}_{initial}$
• Final Momentum: ${\vec{p}}_f=m_1{\vec{v}}_1+m_2{\vec{v}}_2$
• Conservation of Momentum:

$M{\vec{V}}_{initial}=m_1{\vec{v}}_1+m_2{\vec{v}}_2$

The vector sum of the final momenta of the fragments must equal the initial momentum of the shell.

5. Kinetic Energy in an Explosion

Q: Is kinetic energy conserved in an explosion?

A: No. In fact, kinetic energy increases dramatically during an explosion. The initial kinetic energy might be zero, but the final kinetic energy of the flying fragments is large. This new energy comes from the chemical potential energy stored in the explosive material.

6. Rocket Propulsion in Space

Q: How does a rocket engine work in the vacuum of space?

A: A rocket works by the principle of conservation of momentum. It expels hot gas (fuel) at high velocity in one direction. To conserve the total momentum of the rocket-fuel system, the rocket itself must gain an equal and opposite amount of momentum, pushing it forward. It doesn't need air to "push against."