Question

On the basis of newton’s second law of motion proves that the rate of change of momentum of an object is equal to the force applied to it.

Answer 1

Newton's second law states that force applied to a body is directly proportional to the change in momentum.
Suppose an object of mass m is moving along with a straight line with initial velocity u. After applying force F its velocity change to v.
Let p_i is the initial momentum and p_f is the final momentum.
$$\begin{gathered} Rateofchangeofmomentum=\frac{∆p}{t} \\ Here∆p=changeinmomentum(finalmomentum(p_f)-initialmomentum(p_i\left)\right) \\ t=time \end{gathered}$$
$$\begin{gathered} p_f=mv \\ herev=finalvelocity \\ {p}_i=mu \\ hereu=initialvelocity \\ \Rightarrow Rateofchangeofmomentum=\frac{∆p}{t}=\frac{mv-mu}{t} \\ \frac{∆p}{t}=\frac{m(v-u)}{t} \\ Since\frac{(v-u)}{t}=a(accelerationisrateofchangeofvelocity) \\ \Rightarrow \frac{∆p}{t}=ma \\ AccordingtoNewton\text{'}ssecondlawF=ma \\ \Rightarrow \frac{∆p}{t}=F \\ Hencetherateofchangeofmomentumisequaltoforce. \end{gathered}$$