Linear Momentum: Definition, Formula, Examples, Symbol

Types Of Collision

Edited By Vishal kumar | Updated on Jul 02, 2025 07:33 PM IST

Linear momentum, a fundamental concept in physics, plays a crucial role in understanding the motion of objects. It is defined as the product of an object's mass and velocity, indicating how much force is needed to change its motion. There are various types of linear momentum, such as translational, angular, and relativistic, each relevant in different scenarios. For instance, when you push a shopping cart, the cart's linear momentum increases as it accelerates, making it harder to stop. Similarly, understanding linear momentum helps explain everyday phenomena like car collisions, where the momentum of each vehicle determines the impact's outcome. By exploring these types, we gain a deeper insight into the laws of motion that govern our daily lives.

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Linear Momentum
Solved Examples Based on Linear Momentum
Summary

Types Of Collision

Linear Momentum

Linear momentum is a core concept in physics that helps us understand how objects move and interact. Defined as the product of an object's mass and its velocity, linear momentum reflects how much force is required to alter the motion of that object.

The 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 of time.
If a body of mass m is moving with velocity, 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.

As $\mathrm{p}=\mathrm{m}_1 \mathrm{v}_1=\mathrm{m}_2 \mathrm{v}_2=$ constant

$\therefore \frac{V_1}{V_2}=\frac{m_2}{m_1} \Rightarrow V \alpha \frac{1}{m}$

Solved Examples Based on Linear Momentum

Example 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.

1) 10

2) 40

3) 30

4) 15

Solution:

Given : Mass, $(m)=1 \mathrm{KgSpeed},(v)=10 \mathrm{~m} / \mathrm{sSo}$, Linear momentum, $\vec{p}=m \vec{v}$ $\vec{p}=m \vec{v} \Rightarrow \vec{p}=1 \mathrm{~kg} \times 10 \mathrm{~m} / \mathrm{s}=10 \mathrm{~kg} \mathrm{~m} / \mathrm{s}$

Hence the answer is the option (1).

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

1) The momentum of the object is doubled

2) The momentum of the object is quadrupled.

3) The momentum remains the same.

4) The momentum is halved.

Solution :

Given :
Speed, $(v)=2 v$
Mass, $(m)=2 m$
We know,
Momentum, $(p)=m v$
According to the question of mass and speed is doubled. Then, $p=2 m \times 2 v=4 m v=4 p$

Hence the answer is the option (2).

Example 3: A ball of mass 0.20 kg hits a wall at an angle of $45^{\circ}$ with a velocity of $25 \mathrm{~m} / \mathrm{s}$. If 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.

1) $5 \sqrt{2} \mathrm{Ns}$
2) 5 Ns
3) $3 \sqrt{2} \mathrm{~N} \mathrm{~s}$
4) 10 Ns

Solution:

Given
mass, $\mathrm{m}=0.2 \mathrm{~kg}$
initial and final speed, $\mathrm{v}=25 \mathrm{~m} / \mathrm{s}$
The magnitude of Initial linear momentum $\left(\overrightarrow{p_i}\right)$ and final linear momentum $\left(\overrightarrow{p_f}\right)$

$
p=\left|\overrightarrow{p_f}\right|=\left|\overrightarrow{p_i}\right|=m v \Rightarrow p=5 \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}
$
The angle between the initial and final momentum vectors is $90^{\circ}$, as shown in the figure below

Change in linear momentum

$\begin{aligned} & \Delta \vec{p}=\overrightarrow{p_f}-\overrightarrow{p_i} \\ & |\Delta \vec{p}|=\left|\overrightarrow{p_f}-\overrightarrow{p_i}\right| \\ & |\Delta \vec{p}|=\sqrt{p^2+p^2-2 p^2 \cos 90^0} \\ & |\Delta \vec{p}|=\sqrt{2} p \\ & \Rightarrow|\Delta \vec{p}|=5 \sqrt{2} \mathrm{~kg} \cdot \mathrm{m} / \mathrm{s}\end{aligned}$

Hence the answer is the option (1).

Example 4: A ball is thrown vertically up (taken as + z-axis) from the ground. The correct momentum-height (p-h) diagram is :

Solution

As we know,

$
\begin{aligned}
& v^2=u^2-2 g h \\
& v=\sqrt{u^2-2 g h} \\
& |P|=m v \\
& |P|=m \sqrt{u^2-2 g h}
\end{aligned}
$
Therefore, the graph between p and h cannot have a straight line.
During the upward journey as h increases, p decreases, and in the downward journey as h decreases p increases.

Hence the answer is the option (4).

Example 5: An object is thrown vertically upwards. At its maximum height, which of the following quantities becomes zero?

1) Momentum

2) Potential Energy

3) Acceleration

4) Force

Solution:

$\rightarrow$ At maximum height, its velocity becomes zero
$\rightarrow$ Therefore, the momentum of the object becomes zero

Hence the answer is the option (1).

Summary

Linear momentum, defined as the product of an object's mass and velocity, is a crucial concept in physics that helps explain the motion of objects. It is a vector quantity, with its direction aligning with the object's velocity. The momentum of an object determines the force required to change its motion. Key examples include calculating momentum changes in collisions and understanding the effects of doubling an object's mass and velocity.

Frequently Asked Questions (FAQs)

1. What are the three main types of collisions in physics?

The three main types of collisions in physics are elastic collisions, inelastic collisions, and perfectly inelastic collisions. In elastic collisions, both momentum and kinetic energy are conserved. In inelastic collisions, momentum is conserved, but some kinetic energy is lost. In perfectly inelastic collisions, the colliding objects stick together after impact, conserving momentum but losing the maximum amount of kinetic energy.

2. How does the conservation of momentum apply to collisions?

The conservation of momentum applies to all types of collisions. It states that the total momentum of a closed system before a collision is equal to the total momentum after the collision. This principle holds true regardless of whether the collision is elastic, inelastic, or perfectly inelastic. It's a fundamental concept in understanding collision dynamics.

3. What is the difference between an elastic and an inelastic collision?

The main difference between elastic and inelastic collisions is the conservation of kinetic energy. In an elastic collision, both momentum and kinetic energy are conserved. The objects bounce off each other with no loss of kinetic energy. In an inelastic collision, momentum is conserved, but some kinetic energy is converted to other forms, such as heat or sound. The objects may deform or stick together partially.

4. Can you explain what a perfectly inelastic collision is?

A perfectly inelastic collision occurs when two objects collide and stick together, moving as a single unit after the collision. In this type of collision, momentum is conserved, but the maximum amount of kinetic energy is lost compared to other collision types. The lost kinetic energy is typically converted into heat, sound, or deformation of the objects.

5. Why is kinetic energy not conserved in inelastic collisions?

Kinetic energy is not conserved in inelastic collisions because some of the initial kinetic energy is converted into other forms of energy during the collision. This can include heat due to friction, sound energy from the impact, or potential energy stored in the deformation of the colliding objects. The total energy is still conserved, but not all of it remains as kinetic energy after the collision.

6. How do you determine if a collision is elastic or inelastic?

To determine if a collision is elastic or inelastic, you need to examine both momentum and kinetic energy before and after the collision. If both momentum and kinetic energy are conserved (remain the same), the collision is elastic. If only momentum is conserved while kinetic energy decreases, the collision is inelastic. You can calculate these values using the masses and velocities of the objects involved.

7. How do you calculate the final velocities of objects after an elastic collision?

To calculate the final velocities of objects after an elastic collision, you use both the conservation of momentum and the conservation of kinetic energy equations. For two objects in a one-dimensional collision:

8. What is the significance of Newton's cradle in understanding collisions?

Newton's cradle is a device that demonstrates the conservation of momentum and energy in a series of elastic collisions. It typically consists of a row of suspended metal balls. When one ball is lifted and released, it strikes the next ball, transferring its momentum through the row. The last ball then swings out with nearly the same velocity as the first ball. This demonstrates how momentum and energy are transferred through a series of collisions without significant loss, illustrating the principles of elastic collisions.

9. What role does the coefficient of restitution play in collisions?

The coefficient of restitution (e) is a measure of the "bounciness" of a collision. It ranges from 0 to 1 and helps characterize the type of collision. A coefficient of 1 indicates a perfectly elastic collision, where kinetic energy is fully conserved. A coefficient of 0 represents a perfectly inelastic collision, where the objects stick together. Values between 0 and 1 indicate partially inelastic collisions. The coefficient is calculated as the ratio of the relative velocity of separation to the relative velocity of approach.

10. Can kinetic energy increase after a collision?

In a closed system, the total kinetic energy cannot increase after a collision due to the law of conservation of energy. However, in certain scenarios involving external forces or energy sources, it might appear that kinetic energy increases. For example, in an explosion (which can be considered a type of collision), chemical potential energy is converted into kinetic energy, increasing the total kinetic energy of the system. But this is not a true increase in the total energy of the system.

11. How do collisions at the atomic or subatomic level differ from macroscopic collisions?

Collisions at the atomic or subatomic level differ significantly from macroscopic collisions due to quantum mechanical effects. At this scale, particles can behave like waves, and their interactions are governed by probability rather than deterministic classical mechanics. Quantum collisions can result in phenomena like tunneling, where particles can pass through energy barriers that classical physics would forbid. Additionally, the conservation of other quantum properties, such as spin or charge, becomes important in these microscopic collisions.

12. How do collisions in fluids differ from collisions between solid objects?

Collisions in fluids differ significantly from those between solid objects. In fluids, collisions occur at the molecular level and are much more frequent and complex. These collisions contribute to fluid properties like pressure and viscosity. Unlike solid object collisions, fluid particle collisions are typically elastic at the molecular level. However, on a macroscopic scale, fluid collisions can appear inelastic due to energy dissipation through viscous forces. Understanding fluid collisions is crucial in fields like aerodynamics, hydrodynamics, and thermodynamics.

13. What role do collisions play in chemical reactions?

Collisions are fundamental to chemical reactions in several ways:

14. How does mass affect the outcome of a collision?

Mass plays a crucial role in determining the outcome of a collision. In any collision, objects with larger masses tend to experience less change in velocity compared to objects with smaller masses. This is due to the conservation of momentum. When a less massive object collides with a more massive one, the less massive object will experience a greater change in velocity. This principle is why a car experiences less impact when colliding with a bicycle than vice versa.

15. What happens to the center of mass during a collision?

The center of mass of a system undergoing a collision continues to move at a constant velocity, regardless of the type of collision. This is a consequence of the conservation of momentum. Even if the individual objects change their velocities dramatically during the collision, the motion of the center of mass remains unaffected. This principle holds true for all types of collisions in a closed system.

16. How do collisions in one dimension differ from those in two or three dimensions?

Collisions in one dimension involve objects moving along a single line, typically simplifying the analysis. In these cases, velocities are scalar quantities (positive or negative). Two- and three-dimensional collisions are more complex, involving vector velocities. While the principles of momentum and energy conservation still apply, you must consider the components of velocity in each dimension separately. Two- and three-dimensional collisions often require more advanced mathematical techniques to analyze fully.

17. What is the relationship between force and collision time in impacts?

The relationship between force and collision time is inverse: the shorter the collision time, the greater the force experienced during the impact. This is described by the impulse-momentum theorem, where impulse (force multiplied by time) equals the change in momentum. For a given change in momentum, a shorter collision time results in a larger force. This principle explains why airbags in cars extend the collision time to reduce the force experienced by passengers.

18. What is the role of deformation in collisions?

Deformation plays a crucial role in collisions, especially in inelastic ones. When objects collide, they may temporarily or permanently change shape. This deformation process absorbs energy, converting some of the initial kinetic energy into potential energy stored in the material's structure or into heat. The extent of deformation affects how much energy is lost in the collision. In elastic collisions, any deformation is temporary and reversible, while in inelastic collisions, some or all of the deformation may be permanent.

19. How does the concept of impulse relate to collisions?

Impulse is closely related to collisions as it represents the change in momentum during an impact. Mathematically, impulse is defined as the product of force and the time over which it acts (J = F * Δt). In a collision, the impulse experienced by an object equals its change in momentum. Understanding impulse helps in analyzing collision effects, especially in situations where reducing the force of impact is important, such as in designing safety features for vehicles or sports equipment.

20. Can you explain the concept of coefficient of restitution in more detail?

The coefficient of restitution (e) is a measure of the "bounciness" of a collision, ranging from 0 to 1. It's defined as the ratio of the relative velocity of separation to the relative velocity of approach for two colliding objects. Mathematically, e = -(v2f - v1f) / (v2i - v1i), where v represents velocities, and subscripts 1 and 2 denote the two objects. A coefficient of 1 indicates a perfectly elastic collision, 0 indicates a perfectly inelastic collision, and values between 0 and 1 represent partially inelastic collisions. The coefficient of restitution helps predict the behavior of objects after collision and is crucial in fields like sports physics and materials science.

21. How do collisions affect the internal energy of colliding objects?

Collisions can significantly affect the internal energy of colliding objects, especially in inelastic collisions. During an inelastic collision, some of the initial kinetic energy is converted into internal energy of the objects. This can manifest as an increase in temperature (thermal energy), deformation of the objects (elastic potential energy), or even changes in the objects' molecular or atomic structure. In perfectly inelastic collisions, the maximum amount of kinetic energy is converted to internal energy. Understanding these energy transformations is crucial for analyzing real-world collisions and their effects on materials.

22. What is the significance of the line of impact in collisions?

The line of impact is the line along which the force acts during a collision. It's significant because it determines how the collision affects the objects' motion. In a central collision, where the line of impact passes through both objects' centers of mass, the collision affects only the objects' translational motion. In an oblique collision, where the line of impact doesn't pass through the centers of mass, the collision can induce both translational and rotational motion. Understanding the line of impact is crucial for predicting the behavior of objects after collision, especially in complex scenarios like billiards or vehicle collisions.

23. What is the role of friction in collisions?

Friction plays a significant role in many real-world collisions, especially those involving surfaces in contact. During a collision, friction can affect the objects' motion in several ways:

24. How do collisions relate to the concept of work and energy?

Collisions are intimately related to the concepts of work and energy. During a collision, forces between objects do work, leading to energy transfers and transformations. In elastic collisions, the work done by collision forces is reversible, conserving kinetic energy. In inelastic collisions, some of this work goes into deforming the objects or generating heat, leading to a loss of kinetic energy. The work-energy theorem, which states that the net work done on an object equals its change in kinetic energy, is particularly useful in analyzing collisions and understanding energy transformations during impacts.

25. What is the significance of momentum exchange in collisions?

Momentum exchange is a fundamental aspect of all collisions. It refers to the transfer of momentum between colliding objects. The significance of momentum exchange lies in its universality - it occurs in all types of collisions, whether elastic, inelastic, or perfectly inelastic. The principle of conservation of momentum is based on this exchange. Understanding momentum exchange helps predict the motion of objects after collision, which is crucial in various applications, from particle physics to spacecraft design. It also explains phenomena like Newton's cradle and the behavior of billiard balls after collision.

26. How do collisions at different scales compare, from subatomic to astronomical?

Collisions occur at vastly different scales in the universe, from subatomic to astronomical, and while the fundamental principles of conservation of momentum and energy apply at all scales, the nature and effects of these collisions can vary dramatically:

27. What is the relationship between collision dynamics and the laws of thermodynamics?

Collision dynamics and the laws of thermodynamics are closely related, particularly in the context of energy transformations:

28. How do collisions contribute to pressure in gases?

Collisions play a fundamental role in creating pressure in gases:

29. How are collisions used in particle physics experiments?

Collisions are central to particle physics experiments in several ways: