Conservation of Linear Momentum

Linear Momentum is expressed as the product of the Mass, “m” of an object, and the Velocity, “v” of the object. In case, if an object has high Momentum, then it takes greater effort to bring it to stop.

The formula for Linear Momentum, p is given as:

p = mv

Here, the total Momentum doesn’t get changed. This phenomenon is known as Conservation of Momentum. 

This article depicts more about Conservation of Momentum and Linear Momentum.

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Law of Conservation of Linear Momentum depicts that, “when no external force is applied on the colliding bodies under a given system, then the vector summation of particular bodies of the Linear Momentum neither changes nor is affected by their non-to-one interaction.

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The above figure is regarding the law enabling all to know the behavior of Newton’s cradle.

Linear Momentum Formula

Here below, we will derive the formula for the Linear Momentum of a system of particle

As mentioned above, the Linear Momentum of the particle is:

p = mv

Further, according to Newton’s \[2^{nd}\] law for a single particle, it follows:

F = dp/dt ​

Where,

F = force of the particle.

For “n” number of particles, the total Linear  Momentum depicts as,

\[p = p_{1} + p_{2} + ....+ p_{n}\]

Individual Momentum is written as:

\[m_{1}v_{1} + m_{2}v_{2} + ....+ m_{n}v_{n}\]

Now, the Velocity of the center of mass is given by:

\[V = \sum \frac{m_{i}v_{i}}{M}\]

\[mv = \sum m_{i}v_{i}\]

If we compare the above two equations, we get,

p = mv --------(i)

Thus, it can be said that the total Linear Momentum of a system equals the product of the Velocity of the center of mass and the total mass of the system.

After differentiation of equation (i), we get

dp/dt​ = mdv/dt​ = ma

Where,

dv/dt is the acceleration of the center of Mass,

ma is the force external.

Therefore,

\[dp/dt = F_{ext}\]------(ii)

Equation (ii) shows Newton’s \[2^{nd}\] law to be a system of particles where the external force acting over the system is zero.

So, when  \[ F_{ext} = 0 \]---------------(iii)

Then,

dp/dt = 0

The above equation shows that p = constant.

So, when the force acting on the system is zero, then the total Linear Momentum of the system is either conserved or remains constant.

Here, we have proved the law of Conservation of Linear Momentum of a system of particles.

Principle of Conservation of Linear Momentum

Let us consider equation (iii), from above, where the \[ F_{ext} = 0 \].

Here, no external force acts on the isolated system. Under that case, the rate of change of total Momentum doesn’t change. This means the quantity is said to be constant.

The above explanation is the correct derivation for the principle of Conservation of Linear Momentum.

We can say that no matter the characteristic or property of the interaction that goes with any system, the total Momentum will remain as it is. 

Now, let’s come across some examples to have a better understanding of the concept.

Linear Momentum Examples

Let us consider two balls that collide with each other over a billiard table. Here, the billiard table is frictionless. We can use the Momentum of Conservation principle to draw some conclusions and inferences.

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Applying the Principle of Conservation of Linear Momentum

• We need to consider the object that is included in the system.

• The bodies that are in relative to the system, do identify the external and internal forces.

• Verification of the system’s isolated position.

• It should be ensured that the initial Momentum equals to that of the final Momentum.

 Here, the Momentum is a vector quantity.

Ice Skaters

Let us consider two skaters who started from rest, then pushed off against each other on the ice where there is less friction. Here, the woman’s weight is 54 kg, whereas the man’s weight is 88 kg. The woman moves away with a Velocity of 2.5 m/sec. What is the recoil Velocity of man?

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Now, given

\[M_{1}v_{f1} + m_{2}v_{f2} = 0\]

\[v_{f2} = -\frac{M_{1}v_{f1}}{m_{2}}\]

\[v_{f2} = -\frac{(54 kg) (2.5 m/sec)}{88 kg} = -1.5 m/sec\]

Dimensional Formula of Linear Momentum

Momentum is the measure of the product of Mass and Velocity. It is otherwise the quantity of motion of a body in motion.

Now, Linear Momentum = Mass * Velocity ------(1)

The dimensional formula of Mass and Velocity is given by:

Mass = \[(M^{1} L^{0} T^{0})\] ----------(2)

Velocity = \[(M^{0} L^{1} T^{-1})\] -------------(3)

Substitute equation (2) and (3) in equation (1), we get,

p = mv

Or,

L = \[(M^{1} L^{0} T^{0}) * (M^{0} L^{1} T^{-1}) = (M^{1} L^{1} T^{-1})\]

So, the dimensions of Linear Momentum is represented by, \[(M^{1} L^{1} T^{-1})\]

Subatomic Collisions and Momentum

The Conservation of Momentum principle, while chiefly applied to Newtonian mechanics, is not limited to objects in the physically perceived range (macroscopic objects). It also applies to the atomic and subatomic particles, though for objects moving close to the speed of light or with negligible Mass, derivations may vary. Excluding such particles, most other subatomic particles can be analyzed with this principle.  It helps in our explorations of atomic and subatomic particles under laboratory conditions, where giant machines hurl subatomic particles at one another, causing subatomic collisions and researchers then evaluate the results by assuming Conservation of Momentum (among other things).

On such a microscopic scale,  although the particles and their properties are invisible and apparently immeasurable through the naked eye, it is possible to measure their properties with the help of our modern instruments, and models of these subatomic particles can be successfully constructed to describe the results. Momentum is the universal property of all subatomic particles, including Massless particles such as photons that compose light. Momentum is a property of particles (especially the Massless photon) that hints at some greater aspect that is not fully understood as of now. It indicates that Momentum may have an identity beyond the general understanding of it being related to an object’s Mass and Velocity. 

As we gain further insight, we believe that Momentum relates to wave-like properties of particles and plays a foundational role in what measurements are taken and how we take these measurements. Moreover, one can find that the Conservation of Momentum principle is still valid when considering systems of particles. Today this principle is used to analyze the Masses and other properties of particles that were previously undetected through our simple instruments, such as the nucleus of an atom and the existence of "quarks" (strings of energy particles that are known to glue the protons in the nucleus) that makeup particles of nuclei. 

Experiments have now yielded evidence that quarks make up protons, which were initially found to be scattered high-energy electrons off of protons (nuclei of hydrogen atoms). In many experiments, it was noted that electrons occasionally scattered straight backward in a manner that indicated the presence of some very small and very dense particle, which could be a component of the protons this observation is considered nearly absolute evidence of quarks. The analysis was derived partially on the same Conservation of Momentum principle, surprising us on how it works so well even on such a small scale.

Application for Conservation of Linear Momentum

Some of the most identified applications of Conservation of Linear Momentum are:

• The Launching of Rockets- The rocket fuel burns & pushes the exhaust gas in the downward direction. Because of this, the rocket gets pushed in an upward direction.

• Motorboats- It pushes the water backward and gets pushed forward so as to conserve the Momentum.