Question

Prove that the rate of change of momentum is equal to mass $\times$ acceleration.

Answer 1

Proving the rate of change of momentum is equal to the mass $\times$ acceleration

1. According to the second law of Newton, the rate of change of momentum of an object is directly proportional to the resultant force applied and is in the direction of the resultant force.
2. Consider $m$ is the mass of an object, moving with the initial velocity $u$ and the velocity changes to$v$ in time interval t.
3. From Newton's second law

$$\begin{gathered} F\propto \frac{Changeinlinearmomentum}{timeinterval} \\ F\propto \frac{mv-mu}{t} \end{gathered}$$

Here $mv$ is the final momentum and $mu$ is the initial momentum

$F=k\times \frac{m(v-u)}{t}$ Where k is the constant of proportionality.

Experimentally the values of $k=1$

$F=m\left(\frac{v-u}{t}\right)$ …….(i)

or $F=\frac{dp}{dt}$

And we know that rate of change of momentum with respect to time is called acceleration i.e $\left(a\right)=\frac{v-u}{t}$Here $u$ is initial velocity and $v$ is final velocity and $t$ is time.

Putting the above in equation (i) we get

$F=m\left(\frac{v-u}{t}\right)=m\times a$

Hence, from the above equation, we can say that the rate of change of momentum is equal to mass $\times$ acceleration.