Question

Show that the rate of change of momentum is equal to the product of mass and acceleration.

Answer 1

Hint: Here, we are supposed to derive an expression which gives the rate of change of momentum with respect to time equal to the product of mass of the body and its acceleration. In order to proceed, think of these quantities separately and also think of what these values represent and also take into consideration the definition of momentum as well as acceleration.

Complete step by step answer:
First, let us consider the product of mass and acceleration of the object or body. Let the mass of the body be $m$ and the acceleration be $a$. The product of these will be $m \times a = ma$. This is the most famous second law of motion which says that the force on a body is given as the product of the mass and acceleration. So, we have $F = ma$.

Now, let us consider the momentum of the body. Let the object’s momentum be $p$, the velocity with which it is moving be $v$. Momentum is defined as the product of the mass of a body and the velocity of the body. So, we have the momentum of the body as$p = mv$.
We are supposed to find the rate of change of momentum with respect to time, so let us differentiate the momentum.

Differentiating momentum with respect to time, we get,
$\dfrac{d}{{dt}}\left( p \right) = \dfrac{d}{{dt}}\left( {mv} \right)$.
To proceed further, apply the product rule for differentiation which says, \[\dfrac{d}{{dx}}\left( {uv} \right) = u\dfrac{d}{{dx}}\left( v \right) + v\dfrac{d}{{dx}}\left( u \right)\].
So, $\dfrac{{dp}}{{dt}} = m\dfrac{{dv}}{{dt}} + v\dfrac{{dm}}{{dt}}$.
Now, usually we deal with cases where the mass of the object remains constant, therefore, we have, $\dfrac{{dm}}{{dt}} = 0$. Therefore, $\dfrac{{dp}}{{dt}} = m\dfrac{{dv}}{{dt}}$.Now, come back to the first part. The acceleration is defined as the rate of change of velocity with respect to time. Mathematically, $a = \dfrac{{dv}}{{dt}}$. Substituting this value in the expression for rate of change of momentum, we get,$\dfrac{{dp}}{{dt}} = ma$.

Therefore, the rate of change of momentum is equal to the product of mass and acceleration.

Note:Remember the basic definition of the acceleration and momentum of a moving object. Notice that the rate of change of momentum with respect to time is actually equal to the force acting on the object. Note that we have considered the mass to be constant, else the equation will not be valid. The rate of change of momentum is equal to product of mass and acceleration only when mass is constant.