Linear Momentum and its Conservation
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.
Principle of conservation of linear momentum
It is state that if no external force acts on a system, the momentum of the system remains constant.
Consider a system of two bodies on which no external force act. As such, the system is said to be isolated from the surrounding. The bodies can only mutually interact with each other. Due to the mutual interaction of the bodies, the momentum of the individual bodies may change but the net momentum of the system will remain unchanged. Thus, if $ \displaystyle \overrightarrow{{{{p}_{1}}}}\text{ and }\overrightarrow{{{{p}_{2}}}}$ are momentum of the two-bodies at any instant, then in absence of external force
$ \displaystyle \overrightarrow{{{{p}_{1}}}}+\overrightarrow{{{{p}_{2}}}}=\text{ constant}$
If due to mutual interaction, the momentum of the two bodies becomes $ \displaystyle \overrightarrow{{{{p}_{1}}^{\prime }}}\text{ and }\overrightarrow{{{{p}_{2}}^{\prime }}}$ respectively, then according to principal of conservation of momentum
$ \displaystyle \begin{array}{l}\overrightarrow{{{{p}_{1}}}}+\overrightarrow{{{{p}_{2}}}}=\overrightarrow{{{{p}_{1}}^{\prime }}}+\overrightarrow{{{{p}_{2}}^{\prime }}}\\\Rightarrow {{M}_{1}}\overrightarrow{{{{u}_{1}}}}+{{M}_{2}}\overrightarrow{{{{u}_{2}}}}={{M}_{1}}\overrightarrow{{{{v}_{1}}}}+{{M}_{2}}\overrightarrow{{{{v}_{2}}}}\end{array}$
where $ \displaystyle \overrightarrow{{{{u}_{1}}}}\text{ and }\overrightarrow{{{{u}_{2}}}}$ are initial velocities of the two bodies of masses $ \displaystyle {{M}_{1}}\text{ and }{{M}_{2}}$ and $ \displaystyle \overrightarrow{{{{v}_{1}}}}\text{ and }\overrightarrow{{{{v}_{2}}}}$ are their final velocities.
Therefore, the principle of conservation of linear momentum may also be stated as follows:
For isolated system (a system on which no external force act), the initial momentum of the system is equal to the final momentum of the system.
The principle applies equally well, if the system consist of more than two bodies. If no external force acts on a system of n bodies having momentum $ \displaystyle \overrightarrow{{{{p}_{1}}}},\overrightarrow{{{{p}_{2}}}},\overrightarrow{{{{p}_{3}}}},…\overrightarrow{{{{p}_{n}}}}$, then
$ \displaystyle \overrightarrow{{{{p}_{1}}}}+\overrightarrow{{{{p}_{2}}}}+\overrightarrow{{{{p}_{3}}}}+…+\overrightarrow{{{{p}_{n}}}}=\text{ contant}$
Practical applications of the principle of conservation of momentum
1. When a bullet is fired from a gun, the gun recoils or give a force in backward direction.
Let M be the mass of gun and m the mass of bullet. Initially, both the gun and the bullet are at rest. On firing the gun, suppose that the bullet moves with a velocity $ \displaystyle \overrightarrow{v}\text{ }$ and the gun moves with velocity $ \displaystyle \overrightarrow{V}$.
According to the principle of conservation of the momentum,
total momentum of gun and bullet before firing = total momentum of gun and bullet after firing
$ \displaystyle {\text{i}\text{.e}\text{. }0=M\vec{V}+m\vec{v}}$
$ \displaystyle {\Rightarrow \vec{V}=-\frac{m}{M}\vec{v}}$
The negative signs shows that $ \displaystyle \overrightarrow{v}\text{ and }\overrightarrow{V}$ foreign opposite direction i.e. as the bullet moves forward, the gun will move in backward direction. The backward motion of the gun is called recoil of the gun.
2. While firing a bullet, the gun must be held tight to the shoulder
Otherwise because of recoil velocity of the gun, the shoulder may get hurt. If the gun is held tight to the shoulder, then the body of the man firing the gun recoils along with the gun. As the total mass is quite large, the recoil velocity will be very small and the shoulder of the man will not get hurt.
3. When a man jumps from a boat to the shore, the both slightly moved away from the shore.
Initially, the total momentum of the boat and the man is zero. When the man jumps from the boat to the shore, total momentum of man and the boat will be zero only if the boat moves in opposite direction.
4. Rocket works on the principle of conservation of momentum.
As the fuel in the rocket undergoes combustion, the burnt gases leave the body of the rocket with the large velocity in downward direction and thus provide upward thrust to the rocket. If you assume that the fuel is burnt at a constant rate, then rate of change of momentum of the rocket will be constant. As more and more fuel gets burnt, the mass of the rocket goes on decreasing and it leads to increase of the velocity of the rocket more and more rapidly.
It may be pointed out that rocket propulsion is an application of the principal of conservation of momentum to a situation, in which the mass of the system goes on changing.
5. If an astronaut in open space, away from space safe wants to return to his spaceship, he can do so by throwing something in a direction opposite to that in which the spaceship is moving
When the is not throw some object away from the spaceship, he himself will recoil i.e. will move in opposite direction. Due to this, the astronaut will move towards the spaceship.
6. If someone left on a frictionless floor desire to get out of it, he can do so by blowing air out of his mouth.
For the reason explained above, he will move in a direction opposite to in which air is blown out by him. However, in this case the required velocity will be small and it may take him a long time to get out. If it throws away some heavy object, he can acquire comparatively large recall velocity and can get out soon.
