Question

Why is force is said to be the rate of change of momentum and not change in momentum?

Answer 1

Hint: Here, we have to understand the concept of rate of change of momentum and change in momentum and then define why the force is said to be rate of change of momentum but not change in momentum. First, we have to check the relation between momentum and force then we can be able to define the required answer.

Complete step by step solution:
Let us first understand the difference between rate of change of momentum and change in momentum.
Rate of change of momentum: Momentum is product of mass of the object and the velocity of that object. The rate of change of momentum is obtained when the momentum is differentiated with respect to time. Also, the force is directly proportional to the rate of change of momentum as shown below:
\[p = m.v\] …. \[(1)\]
Where, \[p\] is momentum of the object, \[m\] is mass of the object, \[v\] velocity of the object.
Differentiate equation \[(1)\] with respect to time.
\[ \Rightarrow \dfrac{{dp}}{{dt}} = \dfrac{d}{{dt}}\left( {m.v} \right)\]
\[ \Rightarrow \dfrac{{dp}}{{dt}} = m\dfrac{d}{{dt}}(v)\]
\[ \Rightarrow \dfrac{{dp}}{{dt}} = ma\] …. \[\left( {\because \dfrac{{dv}}{{dt}} = a} \right)\] (\[a\] is the acceleration of that object)
\[ \Rightarrow \dfrac{{dp}}{{dt}} = F\] …. \[\left( {\because F = ma} \right)\]
Thus, the rate of change of momentum is said to be equal to force applied to that object.
Now let us understand the change in momentum.
Change in momentum: It is said that the difference between the final and initial momentum of the body at final and initial velocities of it respectively is called the change in momentum. As shown below:
\[{\text{change in momentum = }}{p_{final}} - {p_{initial}}\]
Thus, we conclude that the force is said to be the rate of change of momentum but not the change in momentum.

Note:
> So, we have differentiated between change in momentum and rate of change of momentum. It is a vector quantity since velocity is the vector quantity.
> Vector is the term having magnitude and direction. Scalar quantity is having only magnitude and no direction.