91Ó°ÊÓ

Momentum is a fundamental concept in physics defined as the product of an object's mass and velocity. This quantity is a vector, meaning it has both magnitude and direction. When dealing with multiple objects in a system, the total momentum is the sum of each object's individual momentum. In a closed system, this total momentum is conserved, meaning it stays the same even if the objects interact by exchanging forces.

In the provided exercise, we start with a nucleus initially at rest, implying its total initial momentum is zero. This implies the total momentum of the disintegrated particles must also sum to zero to satisfy the law of conservation of momentum.

• The momentum of the first particle is calculated as: \[ P_1 = m_1 \times v_1 = 5.0 \times 10^{-27} \text{ kg} \times 6.0 \times 10^{6} \text{ m/s}\]
• The second particle's momentum is: \[ P_2 = m_2 \times v_2 = 8.4 \times 10^{-27} \text{ kg} \times 4.0 \times 10^{6} \text{ m/s}\]

Adding these provides the momentum needed by the third particle to ensure the sum remains zero.

Vector addition is a process used to combine two or more vectors to determine their overall impact, often represented as a single resultant vector. When dealing with momentum in multiple dimensions, each directional component must be considered separately. Such components often include the x and y axes in a 2D plane.

In this exercise, the particles have distinct momentum vectors along the x and y axes:

• The first particle's momentum acts along the positive y-axis.
• The second particle's momentum acts along the positive x-axis.

Vectorially, the total of these momenta must be countered entirely by the third particle’s momentum.
When adding these vectors, we sum their components separately:

• The x-component of the third particle's momentum: \[ P_{3x} = -P_{2x} \]
• The y-component of the third particle's momentum: \[ P_{3y} = -P_{1y} \]

This means its momentum vector is the negative or opposite of the vector sum of the first two particles. This results in a vector lying in the negative x and y direction.

Kinematics deals with the motion of objects without considering the forces that cause the motion. It focuses mainly on displacement, velocity, and acceleration. In problems involving explosion or disintegration, like in this exercise, kinematics allows us to calculate the speed and trajectory of objects.

Knowing the mass of each particle through the conservation of mass \[ m_\text{nucleus} = m_1 + m_2 + m_3 \] and the momentum vectors, enables us to calculate the speed of the third particle by rearranging the momentum formula: \[ v_3 = \frac{P_3}{m_3} \].
The final step is finding the velocity's direction, often achieved through simple trigonometry using the vector components. The angle \( \theta \) is calculated using: \[ \theta = \tan^{-1}\left(\frac{P_{3y}}{P_{3x}}\right) \], and adjusted by 180° if the vector is in the negative quadrant, ensuring that it represents the actual path of travel.