Law of Conservation of Linear Momentum

Law of Conservation of Linear Momentum - Examples, FAQs

Edited By Vishal kumar | Updated on Jul 02, 2025 04:30 PM IST

When two cars collide, the total momentum before and after the collision is the same, even though the cars may change speed or direction. This case can be better understood by studying the Law of Conservation of Linear Momentum in Physics. The below article will help us to understand the Law of Conservation of Linear Momentum with their examples in detail.

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State the Law of Conservation of Linear Momentum
Law of Conservation of Momentum Derivation
Example of Law of Conservation of Linear Momentum
Solved Example Based on Conservation of Linear Momentum

Law of Conservation of Linear Momentum - Examples, FAQs

State the Law of Conservation of Linear Momentum

The law of conservation of linear momentum works on the principle that when no external forces act on a system of two colliding objects, the vector sum of their linear momentum remains constant and is unaffected by their mutual interaction.

"The law of conservation of linear momentum states that if a system's net external force is zero, the system's total momentum remains constant."

S.I Unit of linear momentum is $kg \cdot m / s$ and the direction of linear momentum remains the same as the direction of velocity.

Law of Conservation of Linear Momentum Formula

$p = m \cdot v$
Where:

p: Linear momentum (vector quantity)
$m$ : Mass of the object (scalar quantity)
v: Velocity of the object (vector quantity)

Characteristics of Linear Momentum

Linear momentum of a body is the quantity of motion contained in the body.
It is measured in terms of the force required to stop the body in a unit time.
If a body of mass $m$ is moving with velocity $\vec{v}$ , then its linear momentum $\vec{p}$ is given by $\vec{p} = m \vec{v}$ .
It is a vector quantity and its direction is the same as the direction of the velocity of the body.
S.I. Unit: kg-m/sec
Dimension $- {MLT}^{- 1}$
If two objects of different masses have the same momentum, the lighter body possesses greater velocity.

$\begin{aligned} As p = m_{1} v_{1} = m_{2} v_{2} = constant \\ ∴ \frac{V_{1}}{V_{2}} = \frac{m_{2}}{m_{1}} \Rightarrow V α \frac{1}{m} \end{aligned}$

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Law of Conservation of Momentum Derivation

When object A exerts a force on object B, object B reacts with a force of equal magnitude but in the opposite direction, according to Newton's Third Law of Motion

The principle of conservation of momentum is deduced from Newton's third law of motion

Consider two colliding particles A and B with masses of $m_{1}$ and $m_{2}$ with starting and ultimate velocities of $u_{1}$ and $v_{1}$ for A and $u_{2}$ and $v_{2}$ for B, respectively. The contact time between two particles is denoted by the letter t.

As shown in the figure:

law of Conservation of Linear Momentum

The change in the momentum of particle A is:

$Δ P_{A} = m_{1} (v_{1} - u_{1})$

The change in the momentum of particle B is:

$Δ P_{B} = m_{2} (v_{2} - u_{2})$

Using Newton's Third Law of Motion, we can write:

$F_{B A} = - F_{A B}$

Substituting for the forces:

$F_{B A} = m_{2} a_{2} = \frac{m_{2} (v_{2} - u_{2})}{t}$

$F_{A B} = m_{1} a_{1} = \frac{m_{1} (v_{1} - u_{1})}{t}$

Equating the two:

$\frac{m_{2} (v_{2} - u_{2})}{t} = - \frac{m_{1} (v_{1} - u_{1})}{t}$

Simplifying the equation:

$m_{1} u_{1} + m_{2} u_{2} = m_{1} v_{1} + m_{2} v_{2}$

As a result, if no external force is exerted on the system, the momentum after the collision is equal to the momentum before the collision.

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 $m_{2} v_{2} + m_{2} v_{2}$ represents the total momentum of particles A and B after the collision.

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Example of Law of Conservation of Linear Momentum

A Gun's Recoil: When a bullet is shot from a gun, both the bullet and the gun are initially at rest, with zero total momentum. When a bullet is fired, it accelerates forward. As a result of conservation momentum, the cannon acquires a backward momentum. A bullet with mass m is discharged at a forward velocity of v. The mass M gun achieves a rearward velocity u. The overall momentum before firing a bullet from the gun is zero, and the total momentum after firing is also zero.

Rocket propulsion involves a gas chamber at one end from which gas is expelled at high velocity. The total momentum is zero before the ejection. The rocket obtains a rebound velocity and acceleration in the opposite direction due to the ejection of gas. This occurs as a result of momentum conservation.

Motorboats: To conserve momentum, they push the water backward and forward.

These are some of the examples of the conservation of linear momentum that we see in our day-to-day lives. Now we will see some solved examples of the law of conservation of linear momentum.

Solved Example Based on Conservation of Linear Momentum

Question 1: An object of mass 1 kg travels at a constant speed of 10 m/s. Calculate the linear momentum (in kg-m/s )of the object.

(a) 10

(b) 40

(c) 30

(d) 15

Solution:

Given : Mass, $(m) = 1 Kg$

$S p e e d (v) = 10 m / sSo$ ,

Linear momentum, $\vec{p} = m \vec{v}$

$\vec{p} = m \vec{v} \Rightarrow \vec{p} = 1 kg \times 10 m / s = 10 kg m / s$
Hence, the answer is option (a).

Question 2: If the speed and mass of an object are doubled, which of the following is true?

(a) The momentum of the object is doubled

(b) The momentum of the object is quadrupled.

(c) The momentum remains the same.

(d) The momentum is halved.

Solution:

Given :
Speed, $(v) = 2 v$
Mass, $(m) = 2 m$
We know,
Momentum, $(p) = m v$
According to the question if mass and speed are doubled. Then,

$p = 2 m \times 2 v = 4 m v = 4 p$
Hence, the answer is option (b).

Question 3: A ball of mass 0.20 kg hits a wall at an angle of $45^{\circ}$ with a velocity of $25 m / s$ . Suppose the ball rebounds at $90^{\circ}$ to the direction of incidence with the same speed. Calculate the magnitude of change in momentum of the ball.

(a) $5 \sqrt{2} N s$

(b) 5 Ns

(c) $3 \sqrt{2} N s$

(d) 10 Ns

Solution:

Given-
mass, $m = 0.2 kg$
initial and final speed, $v = 25 m / s$
Magnitude of Initial linear momentum $({\vec{p}}_{i})$ and final linear momentum $(\vec{p_{f}}) -$

$p = | \vec{p_{f}} | = | \vec{p_{i}} | = m v \Rightarrow p = 5 kg \cdot m / s$

Angle between initial and final momemtum vectors is $90^{\circ}$ , as shown in the figure below-

change in momentum

Change in linear momentum-

$\begin{aligned} Δ \vec{p} = \vec{p_{f}} - \vec{p_{i}} \\ | Δ \vec{p} | = | \vec{p_{f}} - \vec{p_{i}} | \\ | Δ \vec{p} | = \sqrt{p^{2} + p^{2} - 2 p^{2} \cos 90^{0}} \\ | Δ \vec{p} | = \sqrt{2} p \\ \Rightarrow | Δ \vec{p} | = 5 \sqrt{2} kg \cdot m / s \end{aligned}$
Hence, the answer is an option (a)

Question 4: An object is thrown vertically upwards. At its maximum height, which of the following quantities becomes zero?

(a) Momentum

(b) Potential Energy

(c) Acceleration

(d) Force

Solution:

At maximum height, its velocity becomes zero. Therefore, the momentum of the object becomes zero

The correct option is (1)

Question 5: A body of mass $M$ at rest explodes into three pieces, in the ratio of masses $1 : 1 : 2$ . Two smaller pieces fly off perpendicular to each other with velocities of $30 {ms}^{- 1}$ and $40 {ms}^{- 1}$ respectively. The velocity of the third piece will be :

(a) $15 {ms}^{- 1}$

(b) $25 {ms}^{- 1}$

(c) $35 {ms}^{- 1}$

(d) $5 {ms}^{- 1}$

Solution:

Conservation of momentum,

$\begin{aligned} \frac{m}{4} \times 30 i + \frac{m}{4} 30 J + \frac{m}{2} \vec{v} = 0 \\ \tilde{v} = - 15 i - 20 J \\ | \tilde{v} | = \sqrt{15^{2} + 20^{2}} = 25 m / s \end{aligned}$
Hence, the Correct answer is Option (b).

Frequently Asked Questions (FAQs)

1. What Does Linear Momentum Mean?

The importance of conservation of the linear momentum in a system or body in motion is that it maintains total momentum and is equal to the product of mass and vector velocity when an external force is applied.

2. Determine the linear momentum conservation unit.

Linear momentum p is expressed symbolically as:

p=mv

Where,

The mass of the system is m, and its velocity is v.

As a result, the S.I unit of momentum is kg.m/sec.

The net external force is equal to the change in momentum of a system divided by the rate of change of time, according to Newton's 2nd law of motion.

3. What is linear momentum?

Linear momentum is equal to the product of a system’s mass multiplied by its velocity:

p = mv

Where p=linear momentum

m=mass

v=Velocity

4. Give examples of momentum conservation.

The law of conservation of momentum applies to all physical processes. Here are several examples:

Collision: The conservation of momentum and energy governs the collision of things.
The momentum of the propellant gas causes the rocket to go in the opposite direction, which is due to the law of conservation of momentum.
When a bullet is ejected from a gun, the gun suffers recoil momentum.

5. Can momentum be conserved in two dimensions?

Yes, momentum can be conserved in two (or three) dimensions. In such cases, we consider the components of momentum along each axis separately. The total momentum along each axis is conserved independently of the other axes.

6. How does the conservation of momentum apply in car crashes?

In car crashes, the total momentum of the system (both cars) before and after the collision remains the same. The momentum of each car changes, but these changes balance each other out. This principle is used in designing safety features like crumple zones, which extend the time of collision and reduce the force experienced by passengers.

7. What's the difference between elastic and inelastic collisions in terms of momentum conservation?

In both elastic and inelastic collisions, linear momentum is conserved. The difference lies in the conservation of kinetic energy. In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, momentum is conserved, but some kinetic energy is converted to other forms of energy, like heat or sound.

8. How does mass affect momentum in collisions?

Mass plays a crucial role in momentum. In a collision between objects of different masses, the object with greater mass will experience less change in velocity, while the object with less mass will experience a greater change in velocity. This ensures that the total momentum remains constant.

9. What role does friction play in momentum conservation?

Friction is an external force that can change the momentum of individual objects within a system. However, if we consider the system to include the objects and the surface causing friction, the total momentum is still conserved. The momentum lost by the moving object is transferred to the surface and the Earth.

10. How does the law of conservation of linear momentum apply to rocket propulsion?

In rocket propulsion, the law of conservation of linear momentum explains how rockets move in space. As the rocket expels gas backwards at high speeds, it gains an equal amount of momentum in the forward direction. This is why rockets can accelerate in the vacuum of space without pushing against anything.

11. How does the conservation of momentum apply in explosions?

In explosions, the conservation of momentum still applies. Before the explosion, the total momentum of the system is zero (assuming it's at rest). After the explosion, the fragments move in different directions, but their total momentum remains zero. The momentum of each fragment is balanced by the momentum of other fragments in opposite directions.

12. How does the concept of momentum conservation apply to the game of billiards?

In billiards, the conservation of momentum is evident when balls collide. The total momentum of the system (all the balls involved) remains constant before and after collisions. This principle determines how the balls move after impact, with factors like spin and friction adding complexity to the system.

13. What happens to momentum during perfectly inelastic collisions?

In perfectly inelastic collisions, the colliding objects stick together after impact. While kinetic energy is not conserved in this type of collision, linear momentum is still conserved. The final momentum of the combined object equals the sum of the initial momenta of the individual objects.

14. Can momentum be conserved in systems with changing mass?

Yes, momentum can be conserved in systems with changing mass, but the analysis becomes more complex. This scenario is described by the rocket equation, where the mass of the rocket changes as it burns fuel. The total momentum of the system (rocket + expelled gases) is still conserved.

15. What is the law of conservation of linear momentum?

The law of conservation of linear momentum states that in a closed system, the total linear momentum remains constant before and after a collision or interaction between objects. This means that momentum is neither created nor destroyed, but can be transferred between objects.

16. Why is momentum conserved during collisions?

Momentum is conserved during collisions because of Newton's Third Law of Motion. When objects collide, they exert equal and opposite forces on each other. These forces act for the same amount of time, resulting in equal and opposite changes in momentum, thus preserving the total momentum of the system.

17. Can momentum be created or destroyed?

No, momentum cannot be created or destroyed in a closed system. It can only be transferred between objects or converted into other forms of energy. This is the essence of the law of conservation of linear momentum.

18. How does the conservation of momentum relate to Newton's laws of motion?

The conservation of momentum is closely related to Newton's laws of motion, particularly the third law. Newton's third law states that for every action, there's an equal and opposite reaction. This principle ensures that the changes in momentum of colliding objects are equal and opposite, leading to the conservation of total momentum.

19. Can momentum be conserved in the presence of external forces?

In the presence of external forces, the total momentum of a system may change. However, if we consider the system to include all objects involved (including the source of the external force), then momentum is still conserved within this larger system.

20. How does the concept of center of mass relate to the conservation of momentum?

The center of mass of a system moves as if all the mass were concentrated at that point and all external forces were applied there. In the absence of external forces, the velocity of the center of mass remains constant, which is another way of expressing the conservation of linear momentum for the entire system.

21. What is the difference between momentum and kinetic energy?

While both momentum and kinetic energy involve mass and velocity, they are different concepts. Momentum (p = mv) is a vector quantity with direction, while kinetic energy (KE = ½mv²) is a scalar quantity. In collisions, momentum is always conserved, but kinetic energy may or may not be conserved depending on the nature of the collision.

22. How does the conservation of momentum apply to recoil in firearms?

Recoil in firearms is a direct application of momentum conservation. As the bullet is fired forward, the gun recoils backward. The momentum of the bullet in one direction is balanced by the momentum of the gun in the opposite direction, ensuring that the total momentum of the system remains constant.

23. How does the conservation of momentum relate to the concept of impulse?

Impulse is defined as the change in momentum. The impulse-momentum theorem states that the impulse applied to an object equals its change in momentum. This relationship demonstrates how force applied over time can change an object's momentum while still adhering to the law of conservation of momentum for the entire system.

24. How does momentum conservation apply to nuclear reactions?

In nuclear reactions, such as fission or fusion, momentum conservation is a fundamental principle. It helps predict the velocities and directions of the products of these reactions. Even when mass is converted to energy (as described by E=mc²), the total momentum of the system is still conserved.

25. How is linear momentum calculated?

Linear momentum is calculated by multiplying an object's mass by its velocity. The formula is p = mv, where p is momentum, m is mass, and v is velocity. It is a vector quantity, meaning it has both magnitude and direction.

26. What is the relationship between impulse and momentum?

Impulse is the change in momentum. It's calculated as the product of force and the time interval over which the force acts. The impulse-momentum theorem states that the impulse applied to an object equals its change in momentum. This relationship is expressed as F∆t = ∆p, where F is force, ∆t is time interval, and ∆p is change in momentum.

27. Can momentum be negative?

Yes, momentum can be negative. The sign of momentum depends on the direction of motion relative to the chosen coordinate system. A negative momentum simply means the object is moving in the negative direction of the coordinate axis.

28. Can momentum be conserved in rotational motion?

Yes, there is an analogous concept called angular momentum that is conserved in rotational motion. Just as linear momentum is conserved in linear motion, angular momentum is conserved in rotational motion in the absence of external torques.

29. What is the relationship between momentum conservation and the concept of center of percussion?

The center of percussion, often referred to as the "sweet spot" in sports equipment like baseball bats or tennis rackets, is related to momentum conservation. When an object is struck at its center of percussion, the impact force is not felt at the point where the object is held, making the hit feel effortless. This phenomenon involves the distribution of momentum throughout the object.

30. How does momentum conservation apply to the formation of weather systems?

Momentum conservation plays a role in the formation and behavior of weather systems. For example, the Coriolis effect, which influences the rotation of large-scale weather patterns, is a consequence of momentum conservation in a rotating reference frame (the Earth). This principle helps meteorologists understand and predict the motion of air masses and storm systems.

31. Can momentum be conserved in relativistic scenarios?

Yes, momentum is conserved in relativistic scenarios, but the definition of momentum changes. In special relativity, momentum is defined as p = γmv, where γ is the Lorentz factor. This relativistic momentum is conserved in high-speed collisions and is crucial for understanding particle physics at high energies.

32. What is the significance of momentum conservation in understanding the behavior of superconductors?

In superconductors, the conservation of momentum plays a crucial role in explaining phenomena like the Meissner effect. The collective motion of Cooper pairs (electron pairs that carry current without resistance) in superconductors can be understood in terms of momentum conservation. This principle helps explain why magnetic fields are expelled from superconductors and why they exhibit zero electrical resistance.

33. How does the conservation of momentum apply to subatomic particles?

The conservation of momentum applies to subatomic particles just as it does to macroscopic objects. In particle physics, this principle is crucial for analyzing collisions and decays of particles in accelerators. It helps predict the trajectories and properties of particles produced in high-energy collisions.

34. Can momentum be transferred between objects of different sizes?

Yes, momentum can be transferred between objects of different sizes. In such transfers, the change in velocity of each object is inversely proportional to its mass. For example, when a large truck collides with a small car, the car experiences a much larger change in velocity due to its smaller mass.

35. How does air resistance affect the conservation of momentum?

Air resistance is an external force that can change the momentum of individual objects. However, if we consider the system to include the object and the surrounding air, the total momentum is still conserved. The momentum lost by the object is transferred to the air molecules.

36. What is the role of momentum conservation in the design of airbags?

Airbags work on the principle of momentum conservation. They extend the time of impact during a collision, reducing the force experienced by the occupant. By increasing the time over which the change in momentum occurs, airbags decrease the force and potential for injury.

37. How does the conservation of momentum apply to jet propulsion?

Jet propulsion works on the principle of momentum conservation. As the jet engine expels hot gases backward at high speed, the aircraft gains an equal amount of momentum in the forward direction. This is similar to how rockets work, but jets also use the surrounding air as part of their propulsion system.

38. How does the concept of momentum conservation apply to sports like baseball or cricket?

In sports like baseball or cricket, momentum conservation explains the behavior of the ball after it's hit by the bat. The momentum of the fast-moving bat is transferred to the ball, causing it to change direction and speed. The total momentum of the bat-ball system before and after the collision remains constant.

39. What role does momentum conservation play in understanding cosmic collisions?

Momentum conservation is crucial in understanding cosmic collisions, such as those between galaxies or in the early universe. It helps astronomers predict the outcomes of these massive-scale interactions and understand the formation and evolution of celestial structures.

40. How does the principle of momentum conservation apply to martial arts techniques?

In martial arts, momentum conservation explains the effectiveness of many techniques. For instance, in a throw, the practitioner redirects the opponent's momentum rather than directly opposing it. This redirection, combined with the practitioner's own momentum, results in the opponent being thrown while conserving the total momentum of the system.

41. Can momentum be transferred through non-contact forces?

Yes, momentum can be transferred through non-contact forces like magnetic or gravitational fields. For example, in the gravitational slingshot maneuver used in space missions, a spacecraft gains momentum by passing close to a planet, with the planet losing an equal amount of momentum. The total momentum of the system remains conserved.

42. What is the significance of momentum conservation in particle accelerators?

Momentum conservation is crucial in particle accelerators. It helps physicists predict and analyze the outcomes of high-energy particle collisions. By studying the momenta of particles before and after collisions, scientists can infer the existence and properties of new particles or confirm theoretical predictions.

43. How does the conservation of momentum relate to Newton's cradle?

Newton's cradle demonstrates momentum conservation beautifully. When one ball swings and strikes the others, it transfers its momentum through the stationary balls to the ball at the other end. The number of balls that swing out on the other side is determined by the conservation of both momentum and energy.

44. Can momentum be conserved in systems with friction?

In systems with friction, the linear momentum of the visible objects may not appear to be conserved. However, if we consider the system to include the objects, the surface causing friction, and the Earth, the total momentum is still conserved. The "lost" momentum is actually transferred to the Earth, which has such a large mass that its velocity change is negligible.

45. How does momentum conservation apply to collisions between molecules in a gas?

In a gas, the collisions between molecules obey the law of conservation of momentum. These collisions are typically elastic, meaning both momentum and kinetic energy are conserved. The statistical behavior of these collisions leads to the macroscopic properties of gases, such as pressure and temperature.

46. What is the role of momentum conservation in understanding the motion of celestial bodies?

Momentum conservation is fundamental in celestial mechanics. It helps explain the stable orbits of planets and moons, the trajectories of comets, and the dynamics of star systems. In the absence of external forces, the total momentum of a celestial system remains constant, governing the motion of its components.

47. How does the concept of momentum conservation apply to fluid dynamics?

In fluid dynamics, momentum conservation is expressed through the Navier-Stokes equations. These equations describe how the momentum of fluid particles changes due to pressure differences, viscosity, and external forces. The principle ensures that any change in the fluid's momentum is accounted for by forces acting on it.

48. Can momentum be conserved in quantum mechanics?

Yes, momentum conservation is a fundamental principle in quantum mechanics. It's expressed through the translation symmetry of space and leads to important results like the uncertainty principle. In quantum systems, momentum is conserved in processes like particle decay and scattering, just as in classical mechanics.

49. How does momentum conservation apply to the physics of car safety features?

Car safety features like crumple zones and airbags work on the principle of momentum conservation. They aim to extend the time over which momentum changes occur during a collision, thereby reducing the force experienced by occupants. This is an application of the impulse-momentum theorem, where a longer time of impact results in a smaller force.

50. How does momentum conservation relate to the concept of mechanical waves?

In mechanical waves, such as sound waves or water waves, momentum conservation governs the transfer of energy through the medium. The oscillations of particles in the medium involve exchanges of momentum, and the overall propagation of the wave can be understood in terms of momentum transfer from one region of the medium to another.

51. What role does momentum conservation play in understanding nuclear fusion in stars?

Momentum conservation is crucial in understanding nuclear fusion in stars. In fusion reactions, the momenta of the reacting nuclei and the resulting products must balance. This principle, along with energy conservation, helps astrophysicists model the processes occurring in stellar cores and predict the life cycles of stars.

52. How does the conservation of momentum apply to the physics of sports like diving or gymnastics?

In sports like diving or gymnastics, athletes use the principle of momentum conservation to perform complex rotations and twists. By changing their body configuration mid-air, they can redistribute their angular momentum, allowing them to control their rotation rate without external torques. This is why divers tuck into a ball to spin faster and extend to slow down.

53. Can momentum be conserved in systems with time-varying forces?

In systems with time-varying forces, the instantaneous momentum may change, but the principle of momentum conservation still holds when considering the entire duration of the force application. The total change in momentum of the system will equal the impulse of the applied force over time.

54. How does momentum conservation relate to the concept of pressure in fluids?

Pressure in fluids is closely related to momentum conservation. At a microscopic level, pressure results from the momentum transfer of fluid molecules colliding with surfaces. The force exerted by a fluid on a surface is the rate of change of momentum of the fluid molecules striking the surface, demonstrating a direct link between pressure and momentum conservation.