Question

derive the relationship between force and rate of change of momentum​

Answer 1

As when a body is having some mass and moving with some acceleration then the product of mass and acceleration is said to be Force.

But, when a body is having some mass and moving with some velocity then it is said to be Momentum.

Rate per change in Momentum is called as Force.

Force = Mass × Acceleration

F = ma

Again,

P = m(v-u/t)

F = ma [ v-u/t = a]

Answer 2

Answer:

Let initial momentum ($p_i$) be mu

Let final momentum ($p_f$) be mv

According to 2nd law of motion

$\frac{p_f - p_i}{t} \propto \: f$

$\implies \: f \propto \frac{mv \: - mu}{t}$

$\implies \: f \propto \frac{m(v - u)}{t}$

$f \propto \: ma \: \: \: \: \: \: ( \frac{v - u}{t } = a )$

To remove the proportionality sign. We would add k as the proportionality constant

$f = kma \\ f = ma \:$

because by the definition of force k = 1