A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nucl

Exercises · 4.17

Q. A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions.

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Concept understanding — Conservation of Momentum

Conservation of Momentum: From Push to Principle

Imagine you're standing on perfectly smooth ice, wearing skates. You're completely still. Now, you push a heavy medicine ball away from you. What happens? You roll backward. The harder you push the ball, the faster you roll back.

That's the core intuition: you can't push something away without being pushed back yourself. The push you give the ball is matched by an equal push on you, in the opposite direction. This isn't a special property of ice or skates — it's a fundamental rule of how forces work in the universe.

The Hidden Quantity That Never Changes

Physicists call the "amount of motion" an object has its momentum. For everyday speeds, momentum is simple:

\vec{p} = m \vec{v}

Where $m$ is mass (how much stuff) and $\vec{v}$ is velocity (speed with direction). Momentum is a vector — it cares about which way you're going.

A truck creeping forward has huge momentum (big mass, small speed). A bullet zipping through air has moderate momentum (tiny mass, huge speed). A parked car has zero momentum (speed is zero).

Now here's the key: in any isolated system (no outside forces), total momentum stays the same. Always. Before, during, and after any interaction.

The Precise Statement

Important

Law of Conservation of Momentum:

In a closed, isolated system (no external forces), the total vector momentum of the system remains constant over time.

Mathematically, for two objects that interact (collide, push apart, explode):

\vec{p}_{1,\text{initial}} + \vec{p}_{2,\text{initial}} = \vec{p}_{1,\text{final}} + \vec{p}_{2,\text{final}}

Or in terms of masses and velocities:

m_1 \vec{u}_1 + m_2 \vec{u}_2 = m_1 \vec{v}_1 + m_2 \vec{v}_2

Where $\vec{u}$ means initial velocity and $\vec{v}$ means final velocity.

Why This Works: Newton's Third Law in Disguise

When you push the medicine ball, your hand exerts a force $\vec{F}$ on the ball. By Newton's Third Law, the ball exerts an equal and opposite force $-\vec{F}$ back on your hand. These forces act for the same time $\Delta t$ .

Force times time equals impulse, which equals change in momentum:

\vec{F} \Delta t = \Delta \vec{p}

For you and the ball:

Ball's momentum change: $+\vec{F} \Delta t$ (ball goes forward)
Your momentum change: $-\vec{F} \Delta t$ (you go backward)

Add them: $+\vec{F} \Delta t + (-\vec{F} \Delta t) = 0$

Total change is zero. Momentum is conserved because forces always come in equal-and-opposite pairs.

Note

This is why a rocket works in the vacuum of space. It throws exhaust backward (one momentum change), and the rocket itself moves forward (equal opposite momentum change). No air needed — just Newton's Third Law and conservation of momentum.

What This Law Does NOT Mean

It does NOT mean individual objects keep constant momentum. Only the total of all objects in the system stays constant. Individual momenta can change wildly.
It does NOT apply if external forces act. If friction, gravity from outside, or a wall stops something, momentum is not conserved for that system. (You can expand the system to include the Earth or the wall, and then momentum is conserved again.)
It does NOT require collisions to be elastic. Even in a messy, sticky, energy-losing collision, momentum is still perfectly conserved. Energy can be lost to heat or deformation, but momentum never disappears.

A Quick Example …

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The key idea is that the total momentum of an isolated system is conserved. Since the parent nucleus is at rest, its initial momentum is zero. After disintegration, the two daughter nuclei must have equal and opposite momenta, which forces them to move in opposite directions.

Why Conservation of Momentum is the Right Tool

When a nucleus disintegrates spontaneously (radioactive decay or fission), no external force acts on the system during the brief disintegration process. The parent nucleus and the two daughter nuclei together form an isolated system. In such a system, the total linear momentum remains constant — this is the law of conservation of momentum.

Think of it this way: if you're standing still on frictionless ice and you throw a heavy ball forward, you'll slide backward. The total momentum before and after the throw is zero. The same physics governs nuclear disintegration.

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

Let's apply this to the problem.

Before disintegration — The parent nucleus is at rest in the laboratory frame. Its velocity is zero, so its momentum is zero.

$\vec{p}_{\text{initial}} = 0$
After disintegration — The nucleus splits into two smaller nuclei. Let their masses be $m_1$ and $m_2$ , and their velocities be $\vec{v}_1$ and $\vec{v}_2$ respectively. The total final momentum is:

$\vec{p}_{\text{final}} = m_1 \vec{v}_1 + m_2 \vec{v}_2$
Apply conservation — Since no external force acts, initial momentum equals final momentum:

$0 = m_1 \vec{v}_1 + m_2 \vec{v}_2$

Rearranging:

$m_1 \vec{v}_1 = - m_2 \vec{v}_2$
Interpret the result — The vector equation $m_1 \vec{v}_1 = - m_2 \vec{v}_2$ tells us two things:
- The momentum vectors are opposite in direction (the negative sign).
- Their magnitudes are equal: $m_1 v_1 = m_2 v_2$ . …

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Q.A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions.

Concept understanding — Conservation of Momentum

Conservation of Momentum: From Push to Principle

The Hidden Quantity That Never Changes

The Precise Statement

Why This Works: Newton's Third Law in Disguise

What This Law Does NOT Mean

A Quick Example …

Why Conservation of Momentum is the Right Tool

Concept: Conservation of Linear Momentum in an Isolated System

Q.A nucleus is at rest in the laboratory frame of reference. Show that if it disintegrates into two smaller nuclei the products must move in opposite directions.

Why Conservation of Momentum is the Right Tool

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