91Ó°ÊÓ

Momentum is the product of mass and velocity. It is a vector quantity, which means it has both magnitude (size) and direction. The momentum vector \(\vec{p}\) captures both these aspects.

In our case, the particle's momentum changes direction but not magnitude. This means the length of the momentum vector remains constant, but its direction shifts by \(30^{\circ}\).

A vector can be represented in different ways. Components along orthogonal axes or an arrow in space. The important thing to note is:

- The momentum vector before collision: \(\vec{p}_{\text{initial}}\)
- The momentum vector after collision: \(\vec{p}_{\text{final}}\)

Since only the direction changes and not the magnitude, \(\vec{p}_{\text{initial}}\) and \(\vec{p}_{\text{final}}\) are otherwise identical in size. Knowing these vectors can help calculate the change in momentum, key for understanding collisions.

An elastic collision is one where both momentum and kinetic energy are conserved.

In such collisions, particles bounce off each other without any loss of speed or deformation. This differs from inelastic collisions, where some energy is converted into other forms like heat or sound.

In the context of our exercise:

• The particle changes direction by \(30^{\circ}\).
• The speed and hence the magnitude of its momentum vector remains the same.

This ensures that the total kinetic energy before and after the collision remains constant.

Understanding elastic collisions can help in various fields like physics, engineering, and even video game design where physical accuracy is essential.

The magnitude of a vector is its length, independent of its direction. For a momentum vector \(\vec{p}\), its magnitude is \(\|\vec{p}\| = p\).

In our problem, we deal with the change in the momentum vector's magnitude resulting from a direction change. Let's break it down:

We've learned that:

• The initial and final momentum vectors have the same magnitude.
• The angle between these vectors is \(30^{\circ}\).

To find the magnitude of the vector representing the change in momentum, we use the formula:

\[\|\vec{p}_{\text{final}} - \vec{p}_{\text{initial}}\| = \sqrt{2p^2 - p^2 \sqrt{3}} = p \sqrt{2 - \sqrt{3}}\]

This accounts for both vectors having the same magnitude and the angle between them. Understanding vector magnitude helps grasp many physical concepts, from simple motion to complex quantum states.