$(a)$ Using Newton's law of motion, derive the relation between force and acceleration.
$(b)$ Define one newton.
$(c)$ Which would require a greater force to accelerate: a $0.5 \, kg$ mass at $5 \, m s^{-2}$ or a $4 \, kg$ mass at $2 \, m s^{-2}$? Give reason.

$(c)$ According to Newton's second law of motion, the rate of change of momentum of an object is proportional to the applied unbalanced force in the direction of the force.
Let a body 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, $F \propto \frac{m(v - u)}{t}$.
Since acceleration $a = \frac{v - u}{t}$, we get $F \propto ma$, or $F = kma$. By defining the unit of force such that $k = 1$, we get $F = ma$.
$(b)$ One newton is defined as the force required to produce an acceleration of $1 \, m s^{-2}$ in an object of mass $1 \, kg$.
$(c)$ For the first case: $F_1 = m_1 \times a_1 = 0.5 \, kg \times 5 \, m s^{-2} = 2.5 \, N$.
For the second case: $F_2 = m_2 \times a_2 = 4 \, kg \times 2 \, m s^{-2} = 8 \, N$.
Since $8 \, N$ > $2.5 \, N$, the $4 \, kg$ mass requires a greater force.