Important Derivations: Newton's Laws of Motion

# Important Derivations: Newton's Laws of Motion | Physics Class 11 - NEET PDF Download

 Table of contents Derivation of First Law of Motion from Newton’s Second Law Law of Conservation of Momentum Derivation Momentum Conservation Principle Derivation of Conservation of Momentum Examples of Law of Conservation of Momentum Solved Problems on Law of Conservation of Momentum Derivation of Centripetal Acceleration Centripetal Acceleration Derivation

## Derivation of First Law of Motion from Newton’s Second Law

To derive the First Law of Motion (Law of Inertia) from Newton’s Second Law of Motion (F = ma), we consider a particle that is at rest or moving with a constant velocity (zero acceleration). The First Law of Motion states that an object remains at rest or moves with a constant velocity unless acted upon by an external force.

Let’s assume the following:
– F = Net external force acting on the particle (resultant force)
– m = Mass of the particle
– a = Acceleration of the particle

According to Newton’s Second Law of Motion, we have:
Now, for the First Law of Motion, we want to consider the case when the particle is at rest or moving with a constant velocity. This means the acceleration (a) is zero. In other words, the net external force acting on the particle is also zero.
This implies that if the net external force acting on the particle is zero (F=0), the acceleration of the particle (a) is also zero. Therefore, the particle will either be at rest or move with a constant velocity.

Hence, we have derived the First Law of Motion (Law of Inertia) from Newton’s Second Law of Motion, which states that an object remains at rest or moves with a constant velocity unless acted upon by an external force.

## 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 Newton’s third law of motion.

## Derivation of Conservation of Momentum

Newton’s 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.
A=m1(v1−u1)

(change in momentum of particle A)
B=m2(v2−u2)
(change in momentum of particle B)
FBA=−FAB
(from third law of motion)

m1u1 + m2u2 = m1v1 + m2v2
Therefore, above is the equation of law of conservation of momentum where
m1u1 + m2u2
is the representation of total momentum of particles A and B before the collision and
m1v1 + m2v2
is the representation of total momentum of particles A and B after the collision.

## 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 4 kg and 10 kg respectively that are at rest. The car having the mass 10 kg moves towards the east with a velocity of 5 m.s-1. Find the velocity of the car with mass 4 kg with respect to ground.
Ans: Given,
m1 = 4 kg
m2 = 10 kg
v1 = ?
v2 = 5 m.s-1
Pinitial = 0, as the cars are at rest
Pfinal = p1 + p2
Pfinal = m1.v1 + m2.v2
= (4 kg).(v1) + (10 kg).(5 m.s-1)
We know from the law of conservation of momentum that,
Pinitial = Pfinal
0=4 kg.v1+50 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 1.5 m.s-1.
Ans: Given,
Mass of bullet, m1 = 5 gram = 0.005 kg
Mass of pistol, m2 = 1.5 kg
The velocity of a bullet, v1 = ?
Recoil velocity of pistol, v2 = 1.5 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
∴ (0.005 kg)(0) + (1.5 kg)(0) = (0.005 kg)(v1) + (1.5 kg)(1.5 m.s-1)
0 = (0.005 kg)(v1)+(2.25 kg.m.s-1)
v1=-450 m.s-1
Hence, the recoil velocity of the pistol is 450 m.s-1.

## Derivation of Centripetal Acceleration

Centripetal acceleration is the rate of change of tangential velocity. The net force causing the centripetal acceleration of an object in a circular motion is defined as centripetal force. The derivation of centripetal acceleration is very important for students who want to learn the concept in-depth. The direction of the centripetal force is towards the centre, which is perpendicular to the velocity of the body.
The centripetal acceleration derivation will help students to retain the concept for a longer period of time. The derivation of centripetal acceleration is given in a detailed manner so that students can understand the topic with ease.
The centripetal force keeps a body constantly moving with the same velocity in a curved path. The mathematical explanation of centripetal acceleration was first provided by Christian Huygens in the year 1659. The derivation of centripetal acceleration is provided below.

## Centripetal Acceleration Derivation

The force of a moving object can be written as
F = ma ........... (1)

From the diagram given above, we can say that,

The triangle PQS and AOB are similar. Therefore,

AB = arc AB = vΔt

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## FAQs on Important Derivations: Newton's Laws of Motion - Physics Class 11 - NEET

 1. What is the derivation of the First Law of Motion from Newton's Second Law?
Ans. The First Law of Motion, also known as the law of inertia, states that an object at rest will remain at rest and an object in motion will continue moving with a constant velocity unless acted upon by an external force. This law can be derived from Newton's Second Law of Motion, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. When the net force acting on an object is zero (ΣF = 0), according to Newton's Second Law, the acceleration is also zero (a = 0). Therefore, the object will either remain at rest or continue moving at a constant velocity, in accordance with the First Law of Motion.
 2. What are Newton's Laws of Motion?
Ans. Newton's Laws of Motion are three fundamental principles that describe the relationship between the motion of an object and the forces acting upon it. The laws are as follows: - Newton's First Law of Motion: An object at rest will remain at rest, and an object in motion will continue moving with a constant velocity unless acted upon by an external force. - Newton's Second Law of Motion: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. - Newton's Third Law of Motion: For every action, there is an equal and opposite reaction.
 3. How is the First Law of Motion related to inertia?
Ans. The First Law of Motion, also known as the law of inertia, is closely related to the concept of inertia. Inertia refers to an object's resistance to changes in its state of motion. According to Newton's First Law, an object at rest will remain at rest (with zero velocity) and an object in motion will continue moving with a constant velocity unless acted upon by an external force. This behavior of objects is a manifestation of their inertia. Objects with more mass have greater inertia and require more force to change their state of motion.
 4. Can the First Law of Motion be applied to everyday situations?
Ans. Yes, the First Law of Motion can be applied to everyday situations. For example: - When a car suddenly stops, the passengers inside tend to continue moving forward due to their inertia until restrained by the seatbelts. - When a tablecloth is pulled quickly from underneath a set of dishes, the dishes tend to remain at rest due to their inertia unless acted upon by friction. - When a ball is thrown upwards, it eventually falls back to the ground due to the gravitational force overcoming its initial upward inertia.
 5. What are some practical applications of Newton's Laws of Motion?
Ans. Newton's Laws of Motion have numerous practical applications in various fields, including: - Engineering and construction: Understanding these laws helps in designing and constructing structures that can withstand forces and maintain stability. - Transportation and vehicle dynamics: The laws are essential in designing vehicles, predicting their motion, and ensuring safety measures. - Sports and athletics: Athletes and coaches utilize these laws to enhance performance and optimize techniques in various sports activities. - Aerospace industry: Newton's Laws are crucial for designing rockets, satellites, and other spacecraft to achieve desired trajectories and overcome gravitational forces. - Biomechanics: These laws are used to study human movement, analyze sports techniques, and design prosthetics and orthopedic devices.

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