Momentum

Any moving object will have the momentum for its motion. Momentum is important as well as the interesting property of a moving body. But this momentum always depends on the mass and velocity of the body. Why? In this topic, we will find the answer to this question. Here we will see the momentum formula with some examples. Linear momentum is one of the main categories of momentum. As it is much common due to simple linear motion.

What is Linear Momentum?

If we are standing at the bottom of a hill and we are faced with the option of stopping a bike or a bicycle, then we will probably choose to stop the bicycle. The reasoning behind this is that the bike has more momentum than the bicycle. Here, momentum simply refers to the mass in a moving body.

The bike will have a large momentum because it is a very massive object. Also, it has more speed, which influences the momentum, as well. The bicycle will also have the momentum due to its speed, but due to its less mass, its momentum is also less. In our daily life, we can see examples such as the use of airbags in automobiles. Airbags are used in vehicles because they are able to minimize the effect of the force on any object which may be involved in the collision. Actually, airbags do this by extending the time required to stop the momentum of the driver.

The Formula for Linear Momentum

Linear momentum is defined as the product of the mass (m) of an object and the velocity (v) of the object. This relationship can be described in the form of an equation. It is given as:

The momentum centre of an object is equivalent to the mass of the object times the velocity of the object.

=> P = MV

Here,

M = Mass of the object

V = Velocity of the object

We know that velocity is the speed with direction. Therefore, if an object has a large speed, it also has a large velocity. 

The units of linear momentum are kg - m per sec.

In vector form, we have

\[\overrightarrow{P}\] = \[\overrightarrow{mv}\] …. (1)

According to Newton’s second law of motion, for a fixed mass particle, we know that,

\[\overrightarrow{F}\] = \[\overrightarrow{ma}\] ….(2)

Also,

So, \[\overrightarrow{a}\] = \[\frac{\overrightarrow{dv}}{dt}\]  …(3)

In equation (2) from equation (3), we have

\[\overrightarrow{F}\] = m \[\frac{\overrightarrow{dv}}{dt}\]

=> \[\frac{d}{dt}\] \[\overrightarrow{(mv)}\]

Thus from equation (1) in equation (4), we have

\[\overrightarrow{F}\] = \[\frac{\overrightarrow{dP}}{dt}\] ….(5)

Thus, equation (5) shows that the rate of change in momentum with respect to time is the force applied to the body.

Note that the momentum will always be in the direction as of the velocity. It is a conserved quantity i.e. the total of the momentum of a system will always be constant. It is the law of conservation of momentum. 

Linear Momentum of a System of Particles

The linear momentum of the system of particles is also applicable and important too. Let us consider that for a system of n particles with masses m1, m2, m3, ...mn and velocities \[\overrightarrow{v_{1}}\], \[\overrightarrow{v_{2}}\], \[\overrightarrow{v_{3}}\],........\[\overrightarrow{v_{n}}\], respectively, then the net linear momentum of the system will be as follows:

(equations to be added soon)

Therefore, net momentum will be the product of Mass of the whole system and velocity of the centre of mass of the system. Thus, here M is the centre of mass, and \[\overrightarrow{v_{cm}}\] is the velocity of its centre of mass.

Linear Momentum and Kinetic Energy

The magnitude of linear momentum can also be expressed in terms of the kinetic energy as well.

As we know,

P = mv

Squaring on both sides, we get 

P2 = m2v2

=> P2= 2m (½ mv2)

Since kinetic energy K is ½ mv2

So,

P = \[\sqrt{2mK}\]

Law of Conservation of Linear Momentum

This law is very important in the study of motion, force, and momentum. If the net force acting on some object is equal to zero, then the momentum of the body will remain constant. This fact is termed as the law of conservation of linear momentum.

Since,

Fnet = 0

So,  d/dt Pnet = 0 (from equation 5).

Therefore, Pnet will not change with time. Thus it will be constant. This is the law of conservation of momentum.

If the velocity of the centre of mass is equal to zero. i.e.vcm = 0

Then Pnet will also be constant for the system of particles. Thus, If the velocity of the centre of mass is constant i.e. vcm  = constant) ,

Then, we will get Pnet ​ = constant.

This is known as the law of conservation of linear momentum applicable to the system of particles. Therefore, when the total force acting on the system of a particle is zero, then the total linear momentum of the system will be constant. 

Fun Facts

• Momentum is a vector quantity and it is pointing in the direction of the velocity.

• A net force produces the change in momentum which is equal to the force multiplied by the time interval during which the force was applied.

• Momentum may be linear as well as angular.

• Objects with spin motion have angular momentum. 

• A change in the value momentum is termed as an impulse.

• The recoil of a gun is a suitable example of the conservation of momentum.