Law of Conservation of Momentum Derivation

The law of conservation of momentum is one of the most prominent laws in physics. The conservation of momentum law principle tells us that the total momentum of a system is always conserved for an isolated system. Let us learn more about the conservation of momentum along with derivation and solved problems.

Momentum Conservation Principle

Law of conservation of momentum states that

For two or more bodies in an isolated system acting upon each other, their total momentum remains constant unless an external force is applied. Therefore, momentum can neither be created nor destroyed.

The principle of conservation of momentum is a direct consequence of Newtonโs third law of motion.

Derivation of Conservation of Momentum

Newtonโs third law states that for a force applied by an object A on object B, object B exerts back an equal force in magnitude, but opposite in direction. This idea was used by Newton to derive the law of conservation of momentum.

Consider two colliding particles A and B whose masses are m1 and m2 with initial and final velocities as u1 and v1 of A and u2 and v2 of B. The time of contact between two particles is given as t.

$$\begin{array}{l}A=m_{1}(v_{1}-u_{1})\end{array}$$
(change in momentum of particle A)

$$\begin{array}{l}B=m_{2}(v_{2}-u_{2})\end{array}$$
(change in momentum of particle B)

$$\begin{array}{l}F_{BA}=-F_{AB}\end{array}$$
(from third law of motion)

$$\begin{array}{l}F_{BA}=m_{2}*a_{2}=\frac{m_{2}(v_{2}-u_{2})}{t}\end{array}$$
$$\begin{array}{l}F_{AB}=m_{1}*a_{1}=\frac{m_{1}(v_{1}-u_{1})}{t}\end{array}$$
$$\begin{array}{l}\frac{m_{2}(v_{2}-u_{2})}{t}=\frac{-m_{1}(v_{1}-u_{1})}{t}\end{array}$$
 $$\begin{array}{l}m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}\end{array}$$

Therefore, above is the equation of law of conservation of momentum where

$$\begin{array}{l}m_{1}u_{1}+m_{2}u_{2}\end{array}$$
is the representation of total momentum of particles A and B before the collision and
$$\begin{array}{l}m_{1}v_{1}+m_{2}v_{2}\end{array}$$
is the representation of total momentum of particles A and B after the collision.

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Examples of Law of Conservation of Momentum

Following are the examples of law of conservation of momentum:

• Air-filled balloons
• System of gun and bullet
• Motion of rockets

Solved Problems on Law of Conservation of Momentum

Q1. There are cars with masses 4ย kg and 10ย kg respectively that are at rest. The car having the mass 10ย kg moves towards the east with a velocity of 5ย m.s-1. Find the velocity of the car with mass 4ย kg with respect to ground.
Ans: Given,

m1 = 4ย kg

m2 = 10ย kg

v1 = ?

v2 = 5ย m.s-1

Pinitial = 0, as the cars are at rest

Pfinal = p1 + p2

Pfinal = m1.v1 + m2.v2

= (4ย kg).(v1) + (10ย kg).(5ย m.s-1)

We know from the law of conservation of momentum that,

Pinitial = Pfinal

0=4ย kg.v1+50ย kg.m.s-1

v1 = 12.5 m.s-1

Q2. Find the velocity of a bullet of mass 5 grams which is fired from a pistol of mass 1.5 kg. The recoil velocity of the pistol is 1.5ย m.s-1.
Ans: Given,

Mass of bullet, m1 = 5 gram = 0.005ย kg

Mass of pistol, m2 = 1.5ย kg

The velocity of a bullet, v1 = ?

Recoil velocity of pistol, v2 = 1.5ย m.s-1

Using law of conservation of momentum,

m1u1 + m2u2 = m1v1 + m2v2

Here, Initial velocity of the bullet, u1 = 0

Initial recoil velocity of a pistol, u2 = 0

โด (0.005ย kg)(0) + (1.5ย kg)(0) = (0.005ย kg)(v1) + (1.5ย kg)(1.5ย m.s-1)

0 = (0.005ย kg)(v1)+(2.25ย kg.m.s-1)

v1=-450ย m.s-1

Hence, the recoil velocity of the pistol is 450ย m.s-1.

What is momentum?

Momentum is the product of mass and velocity. It is the quantity of measure of the mass of the object and its velocity.

Give the formula for law of conservation of momentum

The formula for the law of conservation of momentum is:

$$\begin{array}{l}m_{1}u_{1}+m_{2}u_{2}=m_{1}v_{1}+m_{2}v_{2}\end{array}$$

List some examples of law of conservation of momentum.

Examples of law of conservation of momentum are:
• Motion of rockets
• Air-filled balloons
• System of gun and bullet
• Is momentum a scalar or a vector quantity?

Momentum is a vector quantity since it has both magnitude and direction.

State TRUE or FALSE: As friction increases, momentum decreases.

TRUE
Test your knowledge on Conservation of momentum derivation

1. Md Basirul

Good

2. Thanks so much for the derivation

3. krishna vamsi

Thank you

4. Aryan bagra

Very Nice notes and explain

5. soumya rawat

thank you

thank you very much

7. Very well explained and thanks for giving Questions. This makes it easier 2 understand the concept

8. Thanks for the Explanation

9. Md Raihan

Excellent

10. Riya moral

Thank you so much for this derivation.
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