91Ó°ÊÓ

In physics, **momentum** is a measure of the motion of an object and is calculated by multiplying its mass by its velocity. When an object experiences a force, it often results in a change in its velocity, thereby altering its momentum. This change in momentum is termed **impulse**. The relationship is expressed by the formula:

• \( \text{Impulse} = \text{Change in Momentum} = m \times \text{Change in velocity} \)

A force applied over a certain period can generate a change in velocity, altering the momentum of an object. For the gas molecule in the exercise, it hits and bounces off the wall. This collision changes its momentum due to the reversal in direction of its velocity component that is perpendicular to the wall. The impulse felt by the molecule is thus the result of this change in momentum. It is important to note that only the perpendicular component of velocity to the wall changes, as the parallel component remains the same.

Any motion at an angle can be split into two components using trigonometry: horizontal and vertical directions. However, in this exercise, the directions are crucially determined relative to the wall: **perpendicular and parallel**.

• The **perpendicular component** is how much of the velocity is directed towards or away from the wall, calculated as \( v_y = v \cos \theta \).
• The **parallel component** runs along the wall's surface and is given by \( v_x = v \sin \theta \).

Understanding these components is crucial as only certain changes affect the impulse generated upon collision. In this context, the parallel component remains unchanged upon impact with the wall since no force acts in this direction to alter it. Therefore, the exercise highlights the role of these components in understanding motion and momentum change during collisions.

When the gas molecule collides with the wall, its **perpendicular velocity component** changes direction. Before the collision, the perpendicular component of the molecule's velocity is moving towards the wall with value \( v_y = v \cos \theta \).Upon impact, this component reverses, moving away from the wall with a velocity of \(-v \cos \theta\). This reversal results in a total change in velocity (denoted \(\Delta v_y\)) for the perpendicular component:

• \(\Delta v_y = -v \cos \theta - (v \cos \theta) = -2v \cos \theta\)

The change explains the negative sign since the molecule's direction post-collision is opposite to its initial path. This significant alteration in the perpendicular velocity directly impacts the calculation of impulse, which is based on this change in velocity. Understanding this concept is critical to solving impulse problems involving reflections from walls or surfaces.