Derivation of Law of Conservation of Momentum: Formula and Derivation

The Law of Conservation of Momentum is a fundamental principle in classical mechanics that states the total momentum of a closed system remains constant if no external forces act on it. Momentum, the product of an object's mass and velocity, is a vector quantity, meaning it has both magnitude and direction. The law is derived from Newton’s laws of motion, particularly the second and third laws, and is widely applicable in systems ranging from microscopic particles to celestial bodies.

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This Story also Contains

1. Law of Conservation of Momentum
2. Derivation of Momentum or Law of Conservation of Momentum in One-Dimensional
3. Derivation of the Law of Conservation of Momentum in Two-Dimensional
4. Examples of Conservation of Momentum
5. Exam-wise Weightage For the Derivation of the Law of Conservation of Momentum
6. Approach to Solve Law of Conservation of Momentum Questions
7. Summary

This principle plays a key role in understanding collisions and interactions between objects, making it essential in fields like physics, engineering, and even astrophysics. In this derivation, we will explore how the conservation of momentum arises naturally from Newton’s laws and illustrate its application through various scenarios.

Law of Conservation of Momentum

If there is no external force acting on an isolated system, its overall momentum remains constant. As a result, if a system's total linear momentum remains constant, the resultant force exerted on it is zero. In the absence of external torque, angular momentum is conserved as well.The law of conservation of momentum is derived from Newton's third law of motion.

Conservation of momentum states that the momentum of the system is always conserved, i.e. initial momentum and final momentum of the system are always conserved. We can also say that the total momentum of the system is always constant. The product of an object's velocity and mass is the object's momentum. It's a quantity with a vector. The overall momentum of an isolated system is conserved, according to conservation of momentum, a fundamental law of physics. In other words, if no external force acts on a system of objects, their overall momentum remains constant during any interaction. The vector sum of individual momentum is the overall momentum. In any physical process, momentum is conserved.

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Derive the Mathematical Formula of Conservation of Momentum

Newton's third law states that when object A produces a force on object B, object B responds with a force of the same magnitude but opposite direction. Newton derived the mathematical formula for the conservation of momentum.

Let us consider two moving balls $A$ and $B$ of masses $m_1$ and $m_2$ and having initial velocities $u_1$ and $u_2$ such that $u_2<u_1$.

Suppose the balls collide at some point and there is no external force acting on this system.

Let their final velocities be $v_1$ and $v_2$ respectively.
According to Newton's third law of motion,
Force on ball $B$ due to $A=-$ Force on ball $A$ due to $B$.

Or, $F_{A B}=-F_{B A} \ldots \ldots \ldots \ldots \ldots \ldots(i)$

Total initial momentum before collision $\left(p_i\right)=m_1 u_1+m_2 u_2$.

Total final momentum after collision $\left(p_f\right)=m_1 v_1+m_2 v_2$.
According to Newton's second law,

$F_{B A}=\frac{p_A^{\prime}-p_A}{t}=\frac{m_1 v_1-m_1 u_1}{t} \ldots \ldots \ldots(i i)$

$F_{A B}=\frac{p_B^{\prime}-p_B}{t}=\frac{m_2 v_2-m_2 u_2}{t} \ldots \ldots \ldots(i i i)$

From $(i),(i i)$ and $(i i i)$,

$
\begin{aligned}
& \frac{m_1 v_1-m_1 u_1}{t}=-\frac{m_2 v_2-m_2 u_2}{t} \\
& \Rightarrow m_1 v_1-m_1 u_1=-\left(m_2 v_2-m_2 u_2\right) \\
& \Rightarrow m_1 v_1+m_2 v_2=m_1 u_1+m_2 u_2 \\
& \Rightarrow \text { Final momentum }\left(p_f\right)=\text { Initial momentum }\left(p_i\right)
\end{aligned}
$

This is called the law of conservation of momentum formula.
As a result, the equation of the law of conservation of momentum is as follows: $m_1 u_1+m_2 u_2$ represents the total momentum of particles A and B before the collision, and $m_1 v_1+m_2 v_2$ represents the total momentum of particles A and B after the collision.

Derivation of Momentum or Law of Conservation of Momentum in One-Dimensional

A one-dimensional collision of two objects can be used to explain momentum conservation. Two objects with masses of $m_1$ and $m_2$ collide while moving in a straight line at velocities of $u_1$ and $u_2$, respectively. They gain velocities $v_1$ and $v_2$ in the same direction after colliding.

Before the impact, the total momentum

$$
\mathrm{p}_{\mathrm{i}}=\mathrm{m}_1 \mathrm{u}_1+\mathrm{m}_2 \mathrm{u}_2
$$
After the impact, the total momentum

$$
\mathrm{p}_{\mathrm{f}}=\mathrm{m}_1 \mathrm{v}_1+\mathrm{m}_2 \mathrm{v}_2
$$
Total momentum is conserved if no other force acts on the system of two objects. Therefore,

$$
\begin{aligned}
& \mathrm{p}_{\mathrm{i}}=\mathrm{p}_{\mathrm{f}} \\
& \mathrm{m}_1 \mathrm{u}_1+\mathrm{m}_2 \mathrm{u}_2=\mathrm{m}_1 \mathrm{v}_1+\mathrm{m}_2 \mathrm{v}_2
\end{aligned}
$$

NCERT Physics Notes :

• NCERT Notes class 11 physics
• NCERT Notes class 12 physics
• NCERT Notes for all subject

Derivation of the Law of Conservation of Momentum in Two-Dimensional

The overall momentum is $p_{i x}=p_1=m_1 u_1$ along the X -axis and $p_{\mathrm{iy}}=\mathrm{m}_2 \mathrm{u}_2$ along the Y -axis before the collision. The overall momentum after the collision is $\mathrm{p}_{\mathrm{fx}}=(\mathrm{m}+\mathrm{M}) \mathrm{ucos} \theta$ along the X -axis and $\mathrm{p}_{\mathrm{fy}}=(\mathrm{m}+\mathrm{M}) \mathrm{usin} \theta$ along the Y -axis where $m$ is mass and $(m+M)$ is the resultant mass when particles get trapped inside it

where $m$ is mass and $(m+M)$ is the resultant mass when particles get trapped inside it.

$
\begin{aligned}
& \mathrm{p}_{\mathrm{ix}}=\mathrm{P}_{\mathrm{fx}} \\
& \mathrm{m}_1 \mathrm{v}_1=(\mathrm{m}+\mathrm{M}) \mathrm{ucos} \theta(1)
\end{aligned}
$
$
m_2 v_2=(m+M) u \sin \theta(2)
$
As a result of squaring and adding equations (1) and (2),

$
\begin{aligned}
& \left(\mathrm{m}_1 \mathrm{v}_1\right)^2+\left(\mathrm{m}_2 \mathrm{v}_2\right)^2=(\mathrm{m}+\mathrm{M})^2 \mathrm{u}^2\left(\cos ^2 \theta+\sin ^2 \theta\right) \\
& u=\frac{\sqrt{m_1^2 v_1^2+m_2^2 v_2^2}}{m+M}
\end{aligned}
$

It is the combined object's speed.

$
\tan \theta=\frac{m_2 v_2}{m_1 v_1}
$
This determines the direction of the velocity.

Examples of Conservation of Momentum

There are several examples which make the explanation for the law of conservation of momentum. Some of the most common examples of Conservation of Momentum are

1. Recoil of a Gun

• When a bullet is fired from a gun, the bullet moves forward, and the gun recoils backwards.
• The forward momentum of the bullet is equal and opposite to the backward momentum of the gun, keeping the total momentum conserved.

2. Collision of Billiard Balls

• When one billiard ball strikes another, the momentum of the moving ball is transferred to the second ball.
• The total momentum of both balls before and after the collision remains constant.

3. Rocket Propulsion

• A rocket expels gas backwards at high speed, and as a result, the rocket moves forward.
• The backward momentum of the expelled gas balances the forward momentum of the rocket, ensuring the total momentum of the system is conserved.

Exam-wise Weightage For the Derivation of the Law of Conservation of Momentum

Apert from Derivation of law of conservation of Momentum class 9 we study this formula in our higher class such as class 11 and this is very important for competitive exam like JEE and NEET. Given below table is the exam wise weightage of this concept.

Approach to Solve Law of Conservation of Momentum Questions

1. Understand the Problem: Identify the masses, initial and final velocities of the objects involved.

2. Apply the Momentum Conservation Formula:

$$

m_1 u_1+m_2 u_2=m_1 v_1+m_2 v_2

$$

where $m_1, m_2$ are masses, $u_1, u_2$ are initial velocities, and $v_1, v_2$ are final velocities.

3. Substitute Given Values: Insert the known data into the equation.

4. Solve for the Unknown: Rearrange and calculate the unknown variable (velocity or mass).

5. Check Directions: Ensure velocity directions (positive/negative) are consistent with the problem.

Read also :

• NCERT notes Class 11 Physics Chapter 5 Laws of Motion
• NCERT solutions for Class 11 Physics Chapter 5 Laws of Motion
• NCERT Exemplar Class 11 Physics Solutions Chapter 5 Laws of Motion

Summary

The Law of Conservation of Momentum states that the total momentum of an isolated system remains constant before and after any interaction, such as a collision. In any interaction between two objects, they exert equal and opposite forces on each other, following Newton's Third Law. This results in changes in momentum for each object. However, the increase in momentum of one object is exactly balanced by the decrease in momentum of the other. Therefore, the total momentum of the system as a whole stays the same. This principle holds true as long as there are no external forces acting on the system, ensuring momentum is conserved in the interaction.