Question

# rate of change of linear momentum is directly proportional to:A. ForceB. AccelerationC. MassD. All

Hint: Define linear momentum. Obtain the mathematical expression for linear momentum. Rate of change in linear momentum can be found by dividing the change in linear momentum with the change in timer at which the change is made.

As we discussed above the linear momentum that is
$P=mv$
Let assume two linear momentum that is ${{P}_{1}}$ at the velocity of ${{v}_{1}}$ and ${{P}_{2}}$ at the velocity of ${{v}_{2}}$.
Now, we have to find out the difference between these two linear momentums which is $\Delta P$
So, the difference of linear momentum is
$\Delta P={{P}_{1}}-{{P}_{2}}$
Where ${{P}_{1}}$and ${{P}_{2}}$ are $m{{v}_{1}}$ and $m{{v}_{2}}$ respectively.
As we discuss before,
For finding the rate of change of linear momentum we have to differentiate the$\Delta P$ with respect to the time.
$\dfrac{\Delta P}{\Delta t}=\dfrac{m({{v}_{1}}-{{v}_{2}})}{\Delta t}=\dfrac{m\Delta v}{\Delta t}$
Where v is the velocity of the object and m is the mass of the object.
We know that the rate of change of velocity is called acceleration and denoted as ‘a’.
So, in the above formula, we could change the $\dfrac{m\Delta v}{\Delta t}$ as 'ma'.
The formula 'ma' is looking familiar because we know that this is Newton's second law $F=ma$.
So, the rate of change of linear momentum is equal to the force
$\dfrac{\Delta P}{\Delta t}=\dfrac{m\Delta v}{\Delta t}=ma=F$
Note: In physics, we also study another formula of linear momentum that is related to kinetic energy is ${{K}_{E}}=\dfrac{{{P}^{2}}}{2m}$. We can also get the answer through this formula but it will be very lengthy as compared to the above formula that we use. So, we have to check the option first and as per the option, we have to choose the right formula which will give us the right answer in a very easy way.