$(a)$ State Newton's second law of motion. Express it mathematically and find the $SI$ unit of force from it.
$(b)$ In the diagram given below, if the card is flicked away with a jerk, what will you observe? Explain the reason for this observation.

(N/A) Newton's second law of motion states that the rate of change of momentum of an object is directly proportional to the applied unbalanced force in the direction of the force.
Mathematically, let an object of mass $m$ have an initial velocity $u$ and a final velocity $v$ after time $t$ under the influence of force $F$.
Initial momentum $p_1 = mu$
Final momentum $p_2 = mv$
Change in momentum = $p_2 - p_1 = m(v - u)$
Rate of change of momentum = $\frac{m(v - u)}{t}$
According to the law, $F \propto \frac{m(v - u)}{t}$
$F = k \cdot m \cdot \frac{(v - u)}{t}$
Since $\frac{(v - u)}{t} = a$ (acceleration), we have $F = kma$.
Taking $k = 1$, we get $F = ma$.
The $SI$ unit of force is the newton $(N)$, where $1 \, N = 1 \, kg \cdot m \cdot s^{-2}$.
$(b)$ Observation: The card flies away, and the coin falls into the glass.
Reason: This happens due to the inertia of rest. The coin is initially at rest and tends to remain at rest due to its inertia. When the card is flicked, it moves away, but the coin, due to its inertia, does not move with the card and falls into the glass under the influence of gravity.