Step 1: Understand Newton's First Law of Motion.
Newton's First Law of Motion, also known as the Law of Inertia, states that an object at rest will stay at rest, and an object in motion will stay in motion with the same speed and in the same direction unless acted upon by an unbalanced force.
Step 2: Analyze the scenario of beating a carpet.
Initially, both the carpet and the dust particles on it are at rest.
When the carpet is beaten with a stick, a force is applied to the carpet, causing it to move suddenly.
However, the dust particles, due to their inertia (tendency to resist changes in their state of motion), tend to remain at rest.
Step 3: Relate the scenario to the Law of Inertia.
Because the carpet moves away quickly while the dust particles try to maintain their state of rest, the dust particles detach from the carpet and fall out. This phenomenon is a direct demonstration of inertia, which is explained by Newton's First Law of Motion.
Step 4: Consider other laws of motion (briefly).
Newton's Second Law of Motion relates force, mass, and acceleration ($F=ma$). While force and acceleration are involved, the primary reason for the dust falling out is the inertia of the dust.
Newton's Third Law of Motion states that for every action, there is an equal and opposite reaction. While there are action-reaction pairs when the stick hits the carpet, this law doesn't directly explain why the dust detaches due to the carpet's movement.
Step 5: Conclude the relevant law.
The phenomenon of dust particles coming out when a carpet is beaten is best explained by Newton's First Law of Motion (Law of Inertia).
$$(1) First law of Motion$$