Is there a relationship between impulse and momentum?

Yes, there is a direct and fundamental relationship between impulse and momentum, often referred to as the Impulse-Momentum Theorem.¹

In short: Impulse is equal to the change in momentum.²

1. Defining the Concepts

To see how they connect, it helps to look at their individual definitions:

Momentum (³p): This is the “quantity of motion” an object has.⁴ It depends on how much stuff is moving and how fast it’s going.
- Formula: ⁵p = mv
Impulse (⁶J): This is a measure of how much a force changes an object’s motion over a period of time.⁷
- Formula: J = FΔt

2. The Impulse-Momentum Theorem

The relationship between the two is expressed by the equation:

J = Δp

Or, more specifically:

FΔt = m(v_f – v_i)

This means that if you apply a force (⁸F) to an object for a certain amount of time (⁹Δt), you will cause a change in that object’s momentum (¹⁰Δp).¹¹

How it’s derived

This relationship actually comes directly from Newton’s Second Law (F = ma).13 Since acceleration (a) is the change in velocity over time (Δv/Δt), we can rewrite the law as:

F = mΔv/Δt

Multiplying both sides by Δt gives us:

FΔt = mΔv

The left side is Impulse, and the right side is the Change in Momentum.

3. Real-World Applications

Understanding this relationship is critical for safety and sports:

Follow-through in Sports: A golfer or baseball player “follows through” on their swing to increase the time (Δt) the club/bat is in contact with the ball. This creates a larger impulse, resulting in a greater change in momentum (a faster ball).
Airbags and Crumple Zones: In a car crash, your change in momentum (Δp) is fixed—you are going to stop. By using an airbag, the car increases the time (Δt) it takes for you to stop. Because Δt is larger, the force (F) exerted on your body is significantly reduced, saving lives.

Impulse and Momentum Connection

This video provides a clear breakdown of how the Impulse-Momentum Theorem is derived from Newton’s Second Law and shows how to apply it to physics problems.

What is the Impulse-Momentum Theorem?

The Impulse-Momentum Theorem states that the impulse applied to an object is equal to the change in its momentum.¹

In simpler terms, if you want to change how fast an object is moving (its momentum), you must apply a force over a certain amount of time (impulse).²

1. The Mathematical Formula

The theorem is expressed by the following equation:

J = Δp

When expanded using the definitions of impulse and momentum, it looks like this:

FΔt = m(v_f – v_i)

Variable Key:

J: Impulse (measured in Newton-seconds, N·s)³
F: Average force applied⁴
Δt: Time interval over which the force acts⁵
m: Mass of the object⁶
v_f: Final velocity⁷
v_i: Initial velocity⁸

2. How It Works

The theorem is a direct consequence of Newton’s Second Law (⁹F = ma).¹⁰ Because acceleration is the change in velocity over time (a = Δv/Δt), we can rewrite the law as F = mΔv/Δt. By moving the time variable to the other side, we get the relationship: Force × Time = Mass × Change in Velocity.¹¹

3. Why It Matters (Real-World Examples)

The theorem is most famous for explaining how to manage impact forces during collisions:¹²

Safety Features (Airbags & Crumple Zones): When a car crashes, the change in momentum (Δp) is set—the car is going to stop. By using an airbag, you increase the time (¹³Δt) of the impact.¹⁴ According to the theorem, if the time increases, the force (¹⁵F) must decrease to keep the equation balanced.¹⁶ This saves lives by reducing the physical impact on passengers.¹⁷
Sports (Follow-through): A golfer or baseball player follows through on their swing to keep the club/bat in contact with the ball for as long as possible. Increasing Δt results in a larger change in momentum, making the ball fly faster and further.
Landing Softly: When you jump off a chair, you instinctively bend your knees.¹⁸ This increases the time it takes for your body to come to a stop, which reduces the force felt by your joints.¹⁹

Impulse Momentum Theorem Physics Problems

This video provides a practical walkthrough of the theorem, showing how to calculate average force and contact time using real-world physics problems.

What is the Principle of Conservation of Linear Momentum?

The Principle of Conservation of Linear Momentum is a fundamental law of physics stating that the total momentum of a closed (isolated) system remains constant, provided no external forces act on it.¹

Essentially, if you have a group of objects that only interact with each other, the sum of their momenta will not change, even if they collide or explode.

1. The Mathematical Representation

The total momentum before an event is equal to the total momentum after the event.² For a system of two objects, this is written as:

p_initial = p_final

m₁v_1,i + m₂v_2,i = m₁v_1,f + m₂v_2,f

Key Requirements

For momentum to be conserved, the system must be:

Closed/Isolated: No matter enters or leaves the system.³
Zero Net External Force: Forces like friction or gravity from outside the system must be negligible.⁴

2. Examples in Action

The conservation of momentum explains many common physical phenomena:

Collisions: In a game of pool, when the cue ball hits a stationary ball, the cue ball slows down (loses momentum) while the second ball speeds up (gains momentum).⁵ The total momentum of the two balls combined remains the same.⁶
Recoil: When a cannon fires a ball, the ball flies forward with high momentum. To conserve the system’s initial zero momentum, the cannon must push backward (recoil) with an equal and opposite momentum.⁷
Rocket Propulsion: A rocket moves forward by pushing exhaust gases backward at high speeds. The “backward” momentum of the gas is exactly balanced by the “forward” momentum of the rocket.

3. Relationship to Newton’s Laws

This principle is actually a direct result of Newton’s Third Law (Action and Reaction).⁸ When two objects collide, the force object A exerts on B is equal and opposite to the force object B exerts on A (⁹F_{A → B} = -F_{B → A}).¹⁰

Since they are in contact for the same amount of time, they experience equal and opposite impulses, which means they experience equal and opposite changes in momentum.¹¹ These changes cancel each other out, leaving the total momentum of the system unchanged.¹²

Conservation of linear momentum

This video offers a helpful visual guide and worked-out problems to demonstrate how the conservation of momentum is used to predict the results of collisions.

How are collisions described in one dimension?

In physics, collisions in one dimension (1D) occur when two objects move along the same straight line before and after they hit each other.¹ Because the motion is restricted to one axis, we describe them using positive and negative signs to indicate direction.

These collisions are categorized based on what happens to the system’s Kinetic Energy (²KE).³

1. Types of 1D Collisions

Regardless of the type, momentum is always conserved in an isolated system.⁴

Collision Type	Momentum Conserved?	KE Conserved?	Description
Elastic	Yes	Yes	Objects bounce off each other perfectly with no loss of energy (e.g., subatomic particles).
Inelastic	Yes	No	Objects bounce off but lose some KE to heat, sound, or deformation (e.g., most real-world bounces).
Perfectly Inelastic	Yes	No (Max Loss)	Objects stick together after impact and move as a single mass (e.g., a clay ball hitting a wall).

2. The Governing Equations

To solve these problems, we use two main conservation laws:⁵

Conservation of Momentum

For any 1D collision between two masses (m₁ and m₂):

m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

(Where u = initial velocity and v = final velocity)8

Conservation of Kinetic Energy (Elastic Only)

If the collision is elastic, we also use:

1/2m₁u₁² + 1/2m₂u₂² = 1/2m₁v₁² + 1/2m₂v₂²

3. Notable “Special Cases” in 1D

Equal Masses (Elastic): If two identical objects (like billiard balls) collide elastically and one is at rest, they simply swap velocities.⁹ The moving ball stops, and the stationary ball moves off with the original speed.¹⁰
The “Stick” (Perfectly Inelastic): Since the objects move together at the same final velocity ($v_f$), the equation simplifies to: v_f = (m₁u₁ + m₂u₂)/(m₁ + m₂ )

Collisions in One Dimension

This video provides a deep dive into the math behind elastic and inelastic collisions, including step-by-step solutions for common physics problems.

How are collisions described in two dimensions?

Describing collisions in two dimensions (2D) requires treating momentum as a vector quantity.¹ This means we can’t just add numbers together; we have to account for both the speed and the specific direction (angles) of the objects.

The most common approach is to break the motion into two independent parts: the x-axis (horizontal) and the y-axis (vertical).²

1. The Method of Components

The “Golden Rule” for 2D collisions is that momentum is conserved separately along each axis.³ If no external forces act on the system, the total momentum in the x-direction before the hit must equal the total momentum in the x-direction after, and the same applies to the y-direction.⁴

The Conservation Equations

For two objects (m₁ and m₂) colliding and moving off at angles (θ₁ and θ₂):

X-axis: ⁵m₁v_1ix + m₂v_2ix = m₁v_1fxcos(θ₁) + m₂v_2fxcos(θ₂)
Y-axis: ⁶m₁v_1iy + m₂v_2iy = m₁v_1fysin(θ₁) + m₂v_2fysin(θ₂)

2. Types of 2D Collisions

Just like in 1D, the behavior of kinetic energy defines the type of collision:

Inelastic (Glancing): Objects bounce off at angles.⁷ Momentum is conserved in both x and y, but some kinetic energy is lost to heat or sound.
Perfectly Inelastic: Objects collide and stick together, moving off as a single mass (⁸m₁ + m₂) at a specific angle.⁹
Elastic: Both momentum and total kinetic energy are conserved.¹⁰ A famous rule for equal-mass elastic collisions (like two billiard balls where one is initially at rest) is that they will always move off at 90° to each other.

3. Step-by-Step Problem Solving

Set up a Coordinate System: Choose an x and y axis (usually aligned with the initial path of one object).¹¹
Resolve Vectors: Use ¹²sin and ¹³cos to break initial and final velocities into ¹⁴x and ¹⁵y components.¹⁶
Apply Conservation: Solve the two momentum equations (one for ¹⁷x, one for ¹⁸y) simultaneously.¹⁹
Recombine (if needed): If you need the final “total” velocity, use the Pythagorean Theorem: ²⁰v = sqrt(v_x² + v_y²).²¹

Collisions in 2 Dimensions

This video is helpful because it demonstrates how to split momentum into x and y components and provides worked examples for both elastic and inelastic scenarios.

What is rocket propulsion?

Rocket propulsion is the application of Newton’s Third Law of Motion—action and reaction—to move a vehicle through space or the atmosphere.¹ Unlike a car that pushes against the road or an airplane that pushes against the air, a rocket carries its own “reaction mass” (fuel and oxidizer) and works by throwing that mass backward at incredibly high speeds.²

1. How It Works (The Physics)

The core of rocket science is the Conservation of Momentum.³ In a closed system, total momentum must remain constant.⁴

Combustion: Inside the rocket’s engine, fuel and an oxidizer burn to create high-pressure, high-temperature gas.⁵
Expulsion: The engine forces this gas out of a nozzle at the back.⁶ This gas has mass (m) and a very high backward velocity (v), giving it backward momentum.
Reaction: To keep the total momentum of the “rocket + gas” system at zero (relative to its starting state), the rocket must gain an equal amount of forward momentum.

2. Key Components of Thrust

The force that moves a rocket is called Thrust.⁷ According to the impulse-momentum relationship, thrust depends on two main factors:

Mass Flow Rate: How much fuel is burned and ejected every second.⁸
Exhaust Velocity: How fast that gas is moving when it leaves the nozzle.⁹

Note: A common misconception is that rocket exhaust “pushes” against the ground or the air.¹⁰ In reality, rockets actually work better in the vacuum of space because there is no air pressure to resist the gas escaping the nozzle.¹¹

3. The Rocket Equation

Because a rocket burns its fuel as it flies, its mass is constantly decreasing.¹² This means that even with a constant thrust, the rocket’s acceleration increases over time because it is getting lighter.¹³ This relationship is captured by the Tsiolkovsky Rocket Equation:

Δv = v_eln(m₀/m_f)

Δv: The total change in velocity.¹⁴
v_e: The effective exhaust velocity.¹⁵
m₀: The initial “wet” mass (rocket + full fuel).¹⁶
m_f: The final “dry” mass (rocket after fuel is gone).

4. Types of Propulsion

Chemical: The most common type (e.g., SpaceX Falcon 9), using liquid or solid propellants to create fire and high-pressure gas.¹⁷
Ion Propulsion: Uses electricity to accelerate individual charged particles (ions) to extreme speeds.¹⁸ It produces low thrust but is incredibly efficient for long-distance deep space missions.
Nuclear Thermal: Uses a nuclear reactor to heat a propellant (like hydrogen) to create thrust.¹⁹

Rocket Propulsion Basics

This video explains the fundamental principles of rocket engines, including how internal pressure and nozzle design work together to generate thrust.

Physics

Earth and Atmospheric Sciences