What Is Momentum?
Momentum is a physical quantity that describes how much motion an object has.
It depends on two factors:
- the mass of the object
- the velocity with which it is moving
An object moving fast or having large mass has greater momentum.
Momentum helps us understand why fast or heavy objects are harder to stop.
Definition of Momentum
Momentum of an object is defined as the product of its mass and velocity.
Momentum=mass×velocity
or,
p=mv
where
- = momentum
- = mass of the object
- = velocity of the object
SI Unit of Momentum
The SI unit of momentum is:
kg m s−1
Momentum is a vector quantity, because velocity has both magnitude and direction.
Factors Affecting Momentum
Momentum depends on:
1. Mass
For the same velocity:
- heavier object → more momentum
- lighter object → less momentum
2. Velocity
For the same mass:
- higher velocity → more momentum
- lower velocity → less momentum
If either mass or velocity is zero, momentum becomes zero.
Momentum – Examples
- A fast-moving cricket ball has higher momentum, and therefore it hurts more than a slow-moving one.
- A truck has a large mass, so even when moving slowly, it can cause severe damage during a collision.
- A bullet has a small mass, but when fired, it has a very high velocity and hence significant momentum.
- A moving train is hard to stop because its large mass gives it a very large momentum.
Momentum and Direction
Since velocity has direction, momentum also has direction.
- An object moving to the right has momentum in that direction
- If it moves to the left, momentum is in the opposite direction
- This directional nature is important when studying collisions.
Special Case: Object at Rest
If an object is at rest:
v= 0 ⇒ p = mv = 0
So, an object at rest has zero momentum, regardless of its mass.
Importance of Momentum
Momentum plays a key role in:
- understanding collisions
- studying recoil of guns
- explaining safety features like airbags and seat belts
- deriving Newton’s Second Law of Motion
The concept of momentum helps connect motion with force and time.
Momentum and Newton’s Second Law
Newton’s Second Law provides a deep connection between force and momentum.
According to Newton’s Second Law:
The force acting on an object is equal to the rate of change of its momentum.
Mathematically,
where
- p = momentum
- t = time
If mass remains constant,
This shows that:
- Force causes a change in velocity
- Force causes a change in momentum
Key Insight
- A larger force produces a faster change in momentum
- A smaller force acting over a longer time can produce the same change in momentum
This is why:
- Airbags increase stopping time and reduce injury
- Cricketers pull their hands back while catching a fast ball
Thus, Newton’s Second Law naturally leads to the concepts of impulse, collisions, and conservation of momentum.
Change in Momentum
Momentum of an object depends on its velocity.
If the velocity of an object changes, its momentum also changes.
The change in momentum is defined as the difference between the final momentum and the initial momentum.
Change in momentum = Final momentum − Initial momentum
Δp=mv−mu
where
- = mass of the object
- = initial velocity
- = final velocity
Change in momentum can occur due to:
- change in speed
- change in direction
- or both
Impulse
To understand how force changes momentum, we introduce the concept of impulse.
Impulse is defined as the product of force and the time for which the force acts.
Impulse = Force × Time
J = FΔt
Impulse is equal to the change in momentum of the object.
Impulse = Change in momentum
FΔt = mv − mu
This relation connects force, time, and momentum.
Impulse – Examples
- A cricketer pulls his hands backward while catching a ball to increase the time of contact, reducing the force on his hands.
- Airbags increase the time during which the passenger comes to rest, reducing the impact force.
- A karate player breaks a slab by applying a large force over a very short time.
In all these cases, the change in momentum is the same, but the force changes depending on the time interval.
Collisions
A collision is an event in which two or more objects come into contact with each other for a very short time and exert forces on each other.
During a collision:
- The forces involved are very large
- The time of interaction is very small
Momentum plays a key role in understanding the motion before and after impact
Types of Collisions
- Elastic Collision
- Inelastic Collision
Elastic Collision
- Total momentum is conserved
- Total kinetic energy is also conserved
- Objects bounce off without permanent deformation
Example: Collision between two billiard balls.
Inelastic Collision
- Total momentum is conserved
- Kinetic energy is not conserved
- Some energy is lost as heat, sound, or deformation
Example: A clay ball hitting the ground and sticking to it.
Special Case: Perfectly Inelastic Collision
- Objects stick together after collision
- They move with a common velocity after impact
Example: A bullet embedding into a wooden block.
In all collisions, momentum is conserved provided no external forces act on the system.
Law of Conservation of Momentum
The Law of Conservation of Momentum states:
When no external force acts on a system, the total momentum of the system remains constant.
Explanation of the Law
Consider two objects A and B interacting with each other.
Before interaction, their total momentum is:
pinitial = m1u1 + m2u2
After interaction, their total momentum is:
pfinal =m1v1 + m2v2
According to the law:
m1u1 + m2u2 = m1v1 + m2v2
Thus, momentum is conserved.
Examples of Conservation of Momentum
- Recoil of a gun: When a bullet moves forward, the gun moves backward.
- Collision of two balls: Momentum before collision equals momentum after collision.
- Rocket propulsion: Gases expelled backward give the rocket forward momentum.
Key Points to Remember
- Change in momentum depends on change in velocity
- Impulse is equal to change in momentum
- Increasing time reduces force for the same impulse
- Momentum is conserved in the absence of external forces
Momentum connects motion, force, and time, and plays a central role in understanding collisions and real-life motion.
Glossary of Key Terms
Recap of the Key Terms in Momentum
- Acceleration: The rate of change of velocity with time.
- Change in Momentum: The difference between the final momentum and the initial momentum of an object.
- Collision Time: The very short duration during which two bodies interact in a collision.
- Collision: An interaction between two or more bodies for a short duration, during which momentum may be transferred.
- Elastic Collision: A collision in which both momentum and kinetic energy are conserved.
- External Force: A force acting on a system from outside the system.
- Final Velocity (v): The velocity of an object after an interaction or change.
- Force: An interaction that can change the state of motion or shape of an object.
- Impact Force: The large force that acts for a very short time during a collision.
- Impulse: The product of force and the time for which the force acts.: It is equal to the change in momentum.
- Inelastic Collision: A collision in which momentum is conserved but kinetic energy is not.
- Inertia: The tendency of an object to resist any change in its state of rest or motion.
- Initial Velocity (u): The velocity of an object before an interaction or change.
- Internal Force: Forces acting between objects within the same system.
- Law of Conservation of Momentum: The principle that total momentum of a system remains constant when no external force acts on it.
- Linear Momentum: Momentum associated with motion along a straight line.
- Mass: The amount of matter contained in an object. It measures inertia.
- Momentum: The quantity of motion possessed by a body. It is the product of mass and velocity.
- Motion: The change in position of an object with respect to time.
- Recoil: The backward motion of a body when another body moves forward from it.
- Rest: A state in which an object does not change its position with time.
- SI Unit: The standard unit used internationally for measurement.: Momentum is measured in kg m s⁻¹.
- Speed: Distance travelled per unit time. It has magnitude only.
- Stopping Time: The time taken by an object to come to rest.
- System: A group of objects considered together for analysis.
- Vector Quantity: A physical quantity that has both magnitude and direction.
- Velocity: The speed of an object in a given direction.
Quiz
Recap the concepts you have learnt. Try to answer the questions. You can find the answer to any question by clicking on the icon.
What is momentum?
Momentum is the quantity of motion of a body, equal to the product of its mass and velocity.
Write the formula for momentum.
p=mv
Which two physical quantities determine the momentum of a body?
Mass and velocity.
What is the SI unit of momentum?
kg m s⁻¹.
Is momentum a scalar or vector quantity?
Momentum is a vector quantity.
Why is momentum a vector quantity?
Because it has both magnitude and direction (same as velocity).
What is the momentum of an object at rest?
Zero.
How does mass affect momentum?
Greater mass results in greater momentum for the same velocity.
How does velocity affect momentum?
Greater velocity results in greater momentum for the same mass.
What happens to momentum if mass is doubled while velocity remains constant?
Momentum is doubled.
Can an object have momentum if it is not moving? Explain briefly.
No, because momentum depends on velocity, which is zero at rest.
What is meant by change in momentum?
It is the difference between final momentum and initial momentum.
Write the expression for change in momentum.
Δp=mv−mu
What is impulse?
Impulse is the product of force and the time for which it acts.
Write the formula for impulse.
J = FΔt
How is impulse related to change in momentum?
Impulse is equal to the change in momentum.
Why is impulse important in studying collisions?
Because it explains how force acting over time changes momentum during collisions.
Why does a fast-moving object have greater momentum than a slow-moving one?
Because momentum increases with velocity.
Why does a heavy object have more momentum than a light object moving at the same speed?
Because momentum increases with mass.
Why is a moving train difficult to stop?
Because it has very large momentum due to its large mass.
Why does a cricketer pull his hands backward while catching a ball?
To increase stopping time and reduce the force acting on the hands.
Why are airbags used in vehicles?
To increase the time of impact and reduce the force on passengers.
Why does a gun recoil when a bullet is fired?
To conserve momentum of the gun–bullet system.
What is meant by conservation of momentum?
The total momentum of a system remains constant if no external force acts.
State the law of conservation of momentum.
When no external force acts on a system, its total momentum remains constant.
Under what condition is momentum conserved?
When the net external force on the system is zero.
Why is momentum conserved during a collision?
Because the forces involved are internal and act for equal and opposite durations.
What is meant by a system in momentum problems?
A system is a group of interacting bodies considered together.
What is the difference between internal and external forces?
Internal forces act between bodies within the system, while external forces act from outside the system.
Why is momentum more useful than velocity in studying collisions?
Because momentum is conserved, whereas velocity may change.
How is Newton’s Second Law related to momentum?
Newton’s Second Law states that force is equal to the rate of change of momentum.
Write Newton’s Second Law in momentum form.
F = dp/dt
How does increasing the stopping time reduce the force acting on a body?
For the same change in momentum, a longer time reduces the force.
Give one real-life application of conservation of momentum.
Rocket propulsion.