# Linear Momentum: Relation with 2nd Law of Newton

Abstract

Momentum is the quantity of motion due to its mass and velocity. Mathematically expressed as p=mv with SI unit newton second Ns. When a force acts on a body, it produces an acceleration in the body and will be equal to the rate of change of momentum of the body.

It can be written as F x t=m vf -m vi. The change in momentum amounts to the product of force and the time for which force is applied. This kind of second law is more general than the  F =ma since it can easily be extended to deal with changes as the body accelerates.

The momentum of a system depends upon its mass and velocity. If no unbalanced or net force acts upon a system, then its momentum remains constant. Therefore, this is the conservation of momentum.

##### Definition of Momentum

The momentum of a body is the quantity of motion it has due to its mass and velocity.

We understand the fact that moving item possesses a quality by means of which it puts in a force on anything that tries to stop it. The faster the object is moving, the more difficult is to stop it. Similarly, if two items move at the exact same speed, then it is more difficult to stop the massive of the two.

This quality of the moving body was called the quantity of motion of the body, by Newton. This term is now called linear momentum of the body and is specified by the relation.

##### Mathematical Form

Linear momentum = p = mv

ln this expression, v is the speed of the mass m. Direct momentum is, therefore, a vector quantity and has the direction of speed.

SI Unit

The SI unit of momentum is kg metre – per second (kgms-1). It can likewise be expressed as newton second(Ns).

Further Reading:  Electric Current - SI Unit, Conventional Direction of Flow & More
##### Momentum and Newton’s Second Law of Motion
###### Statement

When a force acts on a body, it produces an acceleration in the body and will be equal to the rate of change of momentum of the body.

###### Explanation

Consider a body of mass m moving with an initial velocity vi. Suppose an external force F acts on it for time t after which velocity ends up being vf. The acceleration produced by this force is given by

a = vf – vi / t

By Newton’s 2nd law of motion acceleration, a is

a = F/m

Comparing the two values of acceleration

F/m= vf – vi / t

Now,

F x t=mvf -mvi

Where

• m vi –> initial momentum
• mvf –> final momentum

The change in momentum amounts to the product of force and the time for which force is applied. This kind of the second law is more general than the type F =ma since it can easily be extended to account for changes as the body accelerates when its mass also alters. For example, as a rocket speeds up, it loses mass because its fuel is burnt and ejected to offer higher thrust.

### MCQs about Linear Momentum

• What is the mathematical expression for linear momentum?
• A) p = ma
• B) p = mv
• C) p = F × t
• D) p = m^2v
• What is the SI unit of momentum?
• A) Newton (N)
• B) Meter per second squared (m/s^2)
• C) Kilogram meter per second (kg·m/s)
• D) Kilogram meter per second squared (kg·m/s^2)
• Which law of motion relates force to the rate of change of momentum?
• A) Newton’s First Law
• B) Newton’s Second Law
• C) Newton’s Third Law
• D) Law of Conservation of Momentum
• According to Newton’s Second Law, what is the relationship between force and acceleration?
• A) F = mv
• B) F = ma
• C) F = pt
• D) F = mv^2
• What does the change in momentum of a body depend on?
• A) Time only
• B) Force only
• C) Mass and velocity
• D) Acceleration only
• What is the direction of momentum?
• A) Opposite to velocity
• B) Opposite to acceleration
• C) Same as velocity
• D) Perpendicular to velocity
• Which term did Newton use for the “quantity of motion” of a body?
• A) Motionality
• B) Velocity
• C) Momentum
• D) Inertia
• What is the formula for acceleration according to Newton’s Second Law?
• A) a = F × t
• B) a = mv
• C) a = vf – vi / t
• D) a = F / m
• What does the conservation of momentum state?
• A) Momentum remains constant if no force is applied.
• B) Momentum increases with time.
• C) Momentum decreases with velocity.
• D) Momentum decreases with mass.
• Which law of motion deals with changes in momentum as a body accelerates?
• A) Newton’s First Law
• B) Newton’s Second Law
• C) Newton’s Third Law
• D) Law of Conservation of Momentum
• What does F x t represent in the context of momentum?
• A) Initial momentum
• B) Final momentum
• C) Change in momentum
• D) Total momentum
• What is the relationship between force, time, and change in momentum?
• A) F = Δp / t
• B) F = Δp × t
• C) F = Δp + t
• D) F = Δp – t
• What is the SI unit of momentum?
• A) Kilogram meter per second (kg·m/s)
• B) Newton per second (N/s)
• C) Newton meter per second squared (N·m/s^2)
• D) Kilogram meter per second squared (kg·m/s^2)
• According to Newton’s Second Law, what is the relationship between force and acceleration?
• A) Directly proportional
• B) Inversely proportional
• C) No relationship
• D) Constant relationship
• Which quantity of motion is specified by the relation p = mv?
• A) Kinetic energy
• B) Potential energy
• C) Linear momentum
• D) Angular momentum
Further Reading:  Bernoulli’s Equation

1. B) p = mv
2. C) Kilogram meter per second (kg·m/s)
3. B) Newton’s Second Law
4. D) F = mv^2
5. C) Mass and velocity
6. C) Same as velocity
7. C) Momentum
8. D) a = F / m
9. A) Momentum remains constant if no force is applied.
10. D) Law of Conservation of Momentum
11. C) Change in momentum
12. A) F = Δp / t
13. A) Kilogram meter per second (kg·m/s)
14. A) Directly proportional
15. C) Linear momentum

### Summary

In summary, linear momentum, defined as the product of an object’s mass and velocity, is a crucial concept in physics, quantifying the motion of an object. Described by the equation , where represents momentum, signifies mass, and denotes velocity, its SI unit is the newton second (Ns).

This momentum is intrinsically linked with Newton’s Second Law of Motion, which states that when an external force acts on an object, it produces an acceleration equal to the rate of change of momentum. This relationship is expressed mathematically as , where represents force, denotes time, is the final velocity, and is the initial velocity.