Question

An atomic nucleus suddenly bursts apart (fissions) into two pieces. Piece \(A\), of mass \(m_A\), travels off to the left with speed \(v_A\). Piece \(B\), of mass \(m_B\), travels off to the right with speed \(v_B\). (a) Use conservation of momentum to solve for \(v_B\) in terms of \(m_A\), \(m_B\), and \(v_A\). (b) Use the results of part (a) to show that \(K_A/K_B\) = \(m_B/m_A\), where \(K_A\) and \(K_B\) are the kinetic energies of the two pieces.

Short answer

(a) \( v_B = -\frac{m_A}{m_B}v_A \); (b) \(K_A/K_B = m_B/m_A\).

Step-by-step solution

Understanding Conservation of Momentum

The law of conservation of momentum states that in an isolated system, the total momentum before an event must equal the total momentum after the event. Initially, the system was at rest, so the total momentum was zero.

Set Up the Momentum Equation

Initialize the equation by setting the total momentum after fission to zero because an initial momentum of zero implies: \[ m_A \cdot v_A + m_B \cdot v_B = 0 \] Consider the directions by assuming positive direction as right. Therefore, piece A will have a negative velocity (left direction) and piece B a positive one (right direction).

Solving for Velocity of Piece B

Rearrange the conservation of momentum equation to find the velocity of piece B:\[ m_A \cdot v_A = - m_B \cdot v_B \]Solving for \(v_B\) gives:\[ v_B = -\frac{m_A}{m_B} \cdot v_A \]

Kinetic Energy Definitions

The kinetic energy \(K\) of an object is given by the formula:\[ K = \frac{1}{2} m v^2 \]Thus, \(K_A = \frac{1}{2} m_A v_A^2\) and \(K_B = \frac{1}{2} m_B v_B^2\).

Substituting for Velocity in Energy Equation

Use the derived expression for \(v_B\) into the equation for \(K_B\):\[ K_B = \frac{1}{2} m_B \left( -\frac{m_A}{m_B}v_A \right)^2 = \frac{1}{2} m_B \left( \frac{m_A^2}{m_B^2}v_A^2 \right) = \frac{1}{2} \frac{m_A^2}{m_B}v_A^2 \]

Determine the Ratio of Kinetic Energies

Use the expressions for \(K_A\) and \(K_B\) to find the ratio \(\frac{K_A}{K_B}\):\[ \frac{K_A}{K_B} = \frac{\frac{1}{2} m_A v_A^2}{\frac{1}{2} \frac{m_A^2}{m_B} v_A^2} = \frac{m_A}{\frac{m_A^2}{m_B}} = \frac{m_B}{m_A} \]

Final Step: Summary

The solution to part (a) is \( v_B = -\frac{m_A}{m_B} \cdot v_A \), showing that the velocity of piece B is dependent on the ratio of the masses and the velocity of piece A. The result of part (b) demonstrates that the ratio of kinetic energies \( \frac{K_A}{K_B} \) equals \( \frac{m_B}{m_A} \).

Key concepts

Kinetic Energy

Kinetic energy is a fundamental concept in physics that describes the energy an object possesses due to its motion. It is calculated using the formula: \[ K = \frac{1}{2} m v^2 \] where \( K \) represents the kinetic energy, \( m \) is the mass of the object, and \( v \) is its velocity. This equation shows that kinetic energy is directly proportional to both the mass of the object and the square of its velocity.
When an object moves, kinetic energy is transferred from or to other forms of energy, depending on the nature of the interaction with other objects.
In the context of atomic nucleus fission, the kinetic energy of the resulting pieces can be used to calculate how energy is shared between them after fission. By knowing their masses and velocities, we can determine their kinetic energies and thereby understand the energy distribution resulting from the nuclear split.

Atomic Nucleus Fission

Atomic nucleus fission is a nuclear reaction in which a heavy atomic nucleus splits into two smaller nuclei, along with several neutrons and a large amount of energy. This process is often used in nuclear reactors and weapons.
During fission, the original nucleus is at a high energy state, which becomes unstable. When it breaks apart, the resulting nuclei and particles carry away this energy.

• The two resulting nuclei are the products of the fission.
• Energy released during fission comes mainly in the form of kinetic energy of the fission products and neutrons.
• This process also releases additional neutrons that can induce further fissions, creating a chain reaction in certain circumstances.

Understanding this process is crucial in both energy production, where controlled fission is used to produce power, and in the study of energy distribution among particles.

Momentum Equation

In physics, momentum refers to the quantity of motion an object has and is defined as the product of its mass and velocity: \[ p = m \cdot v \] where \( p \) is momentum, \( m \) is mass, and \( v \) is velocity. The conservation of momentum principle is critical for solving problems involving collision or separation like fission.
The law states that in closed systems, momentum before an event equals momentum after the event. For an initially at rest nucleus splitting into two pieces, the total momentum after the fission must still sum to zero, implying: \[ m_A \cdot v_A + m_B \cdot v_B = 0 \] This equation shows that the momentum of one piece is equal and opposite to that of the other for the system to remain balanced. Solving for one of the velocities, as shown in the original exercise, helps demonstrate how momentum conservation is applied practically.

Velocity Ratio

When dealing with scenarios involving multiple objects or systems, understanding the velocity ratio between different entities is important. In the context of atomic nucleus fission, the velocity ratio can be derived using the conservation of momentum.
From the momentum equation, \[ v_B = -\frac{m_A}{m_B} \cdot v_A \] we can see how the velocities of the two fission pieces relate to their respective masses. This ratio indicates that the velocity of Piece B is inversely proportional to the mass ratio between Piece A and Piece B.

• If Piece A is heavier, its slower velocity translates into a faster velocity for Piece B.
• The negative sign indicates that the two pieces move in opposite directions, which is fundamental in maintaining the system's balance.

Understanding these relationships allows us to predict and calculate motion behavior more effectively, especially in complex systems like fission reactions.