Principle of Conservation of Linear Momentum

What is linear momentum?

Linear momentum is defined as the total quantity of motion contained in a body and is measured as the product of the mass of the body and its velocity.

As the quantity of motion in a body can be produced or destroyed by the application of force on it, the body’s momentum is measured by the force required to stop the body in unit time. The force required to stop a moving body depends upon:

1. Mass of the body

When a ball and a big piece of a stone are allowed to fall from the same height, we find that a much greater force is required to stop the big piece of stone then the ball. Thus, larger the mass of a body, the greater is its linear momentum.

2. Velocity of the body

A bullet thrown with the hand can be stopped much more easily than the same Bullet fired from a gun. This is because in the latter case, velocity is much larger. Therefore larger the velocity of a body, the greater is its linear momentum.

Linear momentum formula

The linear momentum of a body depends upon its mass and velocity. It is measured by the product of the mass of the body and its velocity i.e.

$ \displaystyle \text{Momentum = mass }\times \text{ velocity}$

If a body of mass $ \displaystyle m$ is moving with a velocity $ \displaystyle \overrightarrow{v}$, its linear momentum $ \displaystyle \overrightarrow{p}$ is given by

$ \displaystyle \overrightarrow{p}=m\overrightarrow{v}$

Therefore, the formula of linear momentum is given by:

$ \bbox[15px, #e4e4e4, border: 2px solid #000000]{\boldsymbol{\displaystyle \overrightarrow{p}=m\overrightarrow{v}}}$

Linear momentum is a vector quantity. It’s direction is the same as the direction of the velocity of the body.

The SI unit of linear momentum is kg meter per second (kg m/s) and the CGS unit of linear momentum is gram centimeter per second (g cm/s)

The dimensional formula of momentum is $\displaystyle [{{M}^{1}}{{L}^{1}}{{T}^{{-1}}}]$

Suppose that a ball of mass $ \displaystyle {{M}_{1}}$ and a car of mass $ \displaystyle {{M}_{2}}$ ( ) are moving with the same velocity $ \displaystyle v$. If $ \displaystyle {{p}_{1}}$ and $ \displaystyle {{p}_{2}}$ are the momentum of ball and car respectively, then

$ \displaystyle \dfrac{{{{p}_{1}}}}{{{{p}_{2}}}}=\dfrac{{{{M}_{1}}v}}{{{{M}_{2}}v}}\text{ or }\dfrac{{{{p}_{1}}}}{{{{p}_{2}}}}=\dfrac{{{{M}_{1}}}}{{{{M}_{2}}}}$

As $ \displaystyle {{M}_{1}}>{{M}_{2}}$ it follows that $ \displaystyle {{p}_{2}}>{{p}_{1}}$ i.e. if a ball and a car a travelling with same velocity, the momentum of car will be greater than that of the ball. Similarly, we can show that if two objects of same masses are thrown at different velocities, the one moving with greater velocity will possess greater momentum. Finally, if two objects of masses $ \displaystyle {{M}_{1}}$ and $ \displaystyle {{M}_{2}}$ moving with velocities $ \displaystyle {{v}_{1}}$ and $ \displaystyle {{v}_{2}}$ possess equal momentum, then

$ \displaystyle {{M}_{1}}{{v}_{1}}={{M}_{2}}{{v}_{2}}\text{ or }\dfrac{{{{v}_{1}}}}{{{{v}_{2}}}}=\dfrac{{{{M}_{2}}}}{{{{M}_{1}}}}$

In case, $ \displaystyle {{M}_{2}}>{{M}_{1}}$, then $ \displaystyle {{v}_{2}}<{{v}_{1}}$ i.e. if two bodies of different masses possess same momentum, the lighter body possesses greater velocity.

The concept of momentum was introduced by Newton in order to measure the quantitative effect of force.