Impulse and Momentum

Impulse and Momentum

Learning Objectives

Define linear momentum and impulse as vector quantities.
Apply the Impulse-Momentum Theorem to solve for force, time, or velocity changes.
State and apply the principle of Conservation of Linear Momentum.
Analyze 1D and 2D collisions, distinguishing between elastic, inelastic, and perfectly inelastic collisions.
Use the coefficient of restitution to quantify the elasticity of collisions.
Analyze variable mass systems using the Tsiolkovsky rocket equation.

In mechanics, understanding how forces act over time is crucial for analyzing impacts, crashes, and explosions. This is the realm of impulse and momentum. While the Work-Energy theorem deals with forces acting over a distance, the Impulse-Momentum theorem deals with forces acting over time. This distinction is critical because in many real-world scenarios—like a car crash, a hammer striking a nail, or a jet engine providing thrust—the force is not constant and acts over a very short duration. We cannot easily measure the force or the distance during the impact, but we can measure the velocities before and after.

Center of Mass Frame

Center of Mass Frame Concepts

Before analyzing collisions, it is helpful to define the center of mass (CM) velocity. For a system of particles, the CM moves with a constant velocity if no external forces act on the system.

Center of Mass Velocity

Calculates the velocity of the center of mass for a system of particles.

\vec{v}_{cm} = \frac{\Sigma m_i \vec{v}_i}{\Sigma m_i} = \frac{\vec{P}_{total}}{M_{total}}

Variables

Symbol	Description	Unit
$\vec{v}_{cm}$	Center of mass velocity	$\text{m/s}$
$m_i$	Mass of individual particle	kg
$\vec{v}_i$	Velocity of individual particle	$\text{m/s}$
$\vec{P}_{total}$	Total momentum of the system	$\text{kg} \cdot \text{m/s}$
$M_{total}$	Total mass of the system	kg

Center of Mass Frame Concepts

In the CM reference frame, the total momentum of the system is always identically zero. This makes analyzing complex collisions (especially in 2D or 3D) mathematically much simpler.

p

Linear Momentum ( $p$ ) Concepts

Momentum is a measure of "how hard it is to stop" an object. It depends on both how massive the object is and how fast it's moving.

Linear Momentum ( $\vec{p}$ )

The product of an object's mass and its velocity. It is a vector quantity, pointing in the same direction as the velocity. The SI unit is $\text{kg} \cdot \text{m/s}$ .

Linear Momentum

Calculates the linear momentum of an object.

\vec{p} = m\vec{v}

Variables

Symbol	Description	Unit
$\vec{p}$	Linear momentum vector	$\text{kg} \cdot \text{m/s}$
$m$	Mass of the object	kg
$\vec{v}$	Velocity vector	$\text{m/s}$

Linear Momentum ( $p$ ) Concepts

Newton actually formulated his Second Law in terms of momentum, not acceleration. The net force on an object is equal to the rate of change of its momentum.

Newton's Second Law in terms of Momentum

Expresses the net force as the rate of change of momentum.

\Sigma \vec{F} = \frac{d\vec{p}}{dt} = \frac{d(m\vec{v})}{dt}

Variables

Symbol	Description	Unit
$\Sigma \vec{F}$	Net force vector	N
$\vec{p}$	Momentum vector	$\text{kg} \cdot \text{m/s}$
$t$	Time	s
$m$	Mass	kg
$\vec{v}$	Velocity vector	$\text{m/s}$

Linear Momentum ( $p$ ) Concepts

If mass is constant, this simplifies to our familiar $F=ma$ . However, this general form is powerful because it allows us to analyze systems where mass is changing, like a rocket burning fuel.

J

Impulse ( $J$ ) Concepts

If momentum is the "amount of motion," then impulse is the "change in motion" caused by a force acting over a time interval.

Impulse ( $\vec{J}$ )

The integral of a force over the time interval during which it acts. It is a vector quantity, pointing in the same direction as the force. The SI unit is $\text{N} \cdot \text{s}$ (which is equivalent to $\text{kg} \cdot \text{m/s}$ ).

Impulse

Calculates the impulse delivered by a variable or constant force.

\vec{J} = \int_{t_i}^{t_f} \vec{F}(t) \, dt

\vec{J} = \vec{F}_{avg} \Delta t

Variables

Symbol	Description	Unit
$\vec{J}$	Impulse vector	$\text{N} \cdot \text{s}$
$\vec{F}(t)$	Force as a function of time	N
$t_i$	Initial time	s
$t_f$	Final time	s
$\vec{F}_{avg}$	Average force	N
$\Delta t$	Time interval	s

Impulse ( $J$ ) Concepts

For a constant force, the calculation simplifies to multiplying the average force by the time interval.

Impulse ( $J$ ) Concepts

Graphically, impulse is the area under a Force vs. Time curve. This is why airbags and crumple zones save lives in car crashes. They increase the time ( $\Delta t$ ) over which a passenger's momentum is reduced to zero. By increasing the time, the average force ( $\vec{F}_{avg}$ ) required to deliver the necessary impulse ( $\vec{J}$ ) decreases significantly.

Interactive Simulation: Impulse and Momentum

Use this impulse model to see how the same change in momentum can be produced by different force-time combinations.

Interactive Physics Simulation

Impulse-Momentum Collision Pulse Simulator

Compare different collision force profiles (Rectangular, Triangular, or realistic Sinusoidal). Scrub through the motion to see how the area under the force-time curve translates to momentum transfer.

Force Pulse Profile (Shape)

Peak Impact Force900 N

Contact Duration (Δt)0.18 s

Cart Mass3.0 kg

Motion Scrubber0.65

Governing Formulas

Impulse-Momentum Theorem

J = \int F dt = m \cdot \Delta v

Impulse (Area under curve)

J = \frac{1}{2} F_{peak} \cdot \Delta t

Total Impulse (J)

81.0 N·s

Velocity Increase (Δv)

27.00 m/s

The Impulse-Momentum Theorem

The Impulse-Momentum Theorem Concepts

Just as the Work-Energy theorem connects work and kinetic energy, the Impulse-Momentum theorem connects impulse and momentum.

The Impulse-Momentum Theorem

The net impulse exerted on a particle is equal to the change in the particle's momentum.

The Impulse-Momentum Theorem

Relates the net impulse applied to an object to its change in momentum.

\vec{J}_{net} = \Delta \vec{p} = \vec{p}_f - \vec{p}_i = m\vec{v}_f - m\vec{v}_i

Variables

Symbol	Description	Unit
$\vec{J}_{net}$	Net impulse vector	$\text{N} \cdot \text{s}$
$\Delta \vec{p}$	Change in momentum vector	$\text{kg} \cdot \text{m/s}$
$\vec{p}_f$	Final momentum vector	$\text{kg} \cdot \text{m/s}$
$\vec{p}_i$	Initial momentum vector	$\text{kg} \cdot \text{m/s}$
$m$	Mass	kg
$\vec{v}_f$	Final velocity vector	$\text{m/s}$
$\vec{v}_i$	Initial velocity vector	$\text{m/s}$

The Impulse-Momentum Theorem Concepts

This theorem is essentially Newton's Second Law rewritten to focus on the time interval over which a force acts.

Conservation of Linear Momentum

Conservation of Linear Momentum Concepts

The true power of momentum lies in analyzing systems of interacting particles. When two objects collide, the forces they exert on each other are internal to the system. By Newton's Third Law, these forces are equal and opposite, so the impulses they impart to each other are also equal and opposite ( $\vec{J}_{1 \text{ on } 2} = -\vec{J}_{2 \text{ on } 1}$ ).

This leads to a profound conclusion.

Conservation of Linear Momentum

If the net external force on a system is zero ( $\Sigma \vec{F}_{ext} = 0$ ), then the total linear momentum of the system is conserved (remains constant).

Conservation of Linear Momentum

States that total momentum before a collision equals total momentum after a collision for a closed system.

\vec{P}_{total, i} = \vec{P}_{total, f}

m_1\vec{v}_{1i} + m_2\vec{v}_{2i} = m_1\vec{v}_{1f} + m_2\vec{v}_{2f}

Variables

Symbol	Description	Unit
$\vec{P}_{total}$	Total momentum vector	$\text{kg} \cdot \text{m/s}$
$m_1, m_2$	Masses of interacting objects	kg
$\vec{v}_{1i}, \vec{v}_{2i}$	Initial velocity vectors	$\text{m/s}$
$\vec{v}_{1f}, \vec{v}_{2f}$	Final velocity vectors	$\text{m/s}$
$i, f$	Initial and final states	-

Vector Conservation of Momentum

Momentum is a vector quantity. This means momentum must be conserved in each spatial dimension independently. In a 2D collision on an x-y plane:

\begin{aligned} \Sigma p_{xi} &= \Sigma p_{xf} \\ \Sigma p_{yi} &= \Sigma p_{yf} \end{aligned}

Interactive Simulation: Collision Momentum

Use this collision model to see total momentum remain consistent when two carts stick together after impact.

Interactive Physics Simulation

Momentum & Elastic Collision Simulator

Adjust cart masses, incoming velocities, and coefficient of restitution. Scrub through the timeline and watch momentum conservation in action.

Cart A: 2.0kg

Cart B: 3.0kg

Cart A Parameters

Cart A Mass2.0 kg

Cart A Initial Velocity6.0 m/s

Cart B Parameters

Cart B Mass3.0 kg

Cart B Initial Velocity-2.0 m/s

Coefficient of Restitution (e)0.25

Timeline Scrubber0.45

Governing Principles

Conservation of Momentum

m_A v_{Ai} + m_B v_{Bi} = m_A v_{Af} + m_B v_{Bf}

Restitution Coefficient (e)

e = \frac{v_{Bf} - v_{Af}}{v_{Ai} - v_{Bi}}

Momentum before collision (p_i)

6.00 kg·m/s

Momentum after collision (p_f)

6.00 kg·m/s

Velocity Cart A

6.00 m/s

Velocity Cart B

-2.00 m/s

Variable Mass Systems: Rocket Equation

Variable Mass Systems: Rocket Equation Concepts

A classic application of the generalized Newton's Second Law ( $\Sigma \vec{F} = \frac{d\vec{p}}{dt}$ ) is analyzing systems where mass is continuously ejected or accumulated, such as a rocket.

A rocket accelerates by ejecting mass (exhaust gas) backwards at a high relative velocity ( $v_e$ ). Conservation of momentum gives the Tsiolkovsky rocket equation, which relates the change in velocity ( $\Delta v$ ) to the effective exhaust velocity ( $v_e$ ) and the initial ( $m_i$ ) and final ( $m_f$ ) mass of the rocket:

Tsiolkovsky Rocket Equation

Calculates the change in velocity of a rocket based on its exhaust velocity and mass ratio.

\Delta v = v_e \ln\left(\frac{m_i}{m_f}\right)

Variables

Symbol	Description	Unit
$\Delta v$	Change in velocity	$\text{m/s}$
$v_e$	Effective exhaust velocity	$\text{m/s}$
$m_i$	Initial mass (including fuel)	kg
$m_f$	Final mass (after fuel is burned)	kg

Variable Mass Systems: Rocket Equation Concepts

The term $\frac{dm}{dt} v_e$ is known as thrust, the upward force exerted on the rocket by the expelled exhaust.

Types of Collisions

Types of Collisions Concepts

While total momentum is always conserved in a collision (if external forces are negligible), kinetic energy is not necessarily conserved. Collisions are classified by what happens to the total kinetic energy.

Elastic Collisions: Both momentum and total kinetic energy are conserved. The objects bounce off each other perfectly without any loss of mechanical energy to heat or deformation. Example: Billiard balls (approximately), gas molecules.
Inelastic Collisions: Momentum is conserved, but kinetic energy is not. Some kinetic energy is converted into internal energy (heat, sound, deformation). Most real-world macroscopic collisions are inelastic.
Perfectly Inelastic Collisions: A specific type of inelastic collision where the objects stick together and move with a common final velocity after the impact. This results in the maximum possible loss of kinetic energy (though momentum is still conserved).

e

Coefficient of Restitution ( $e$ ) Concepts

The degree of elasticity in a 1D collision is quantified by the coefficient of restitution, $e$ . It relates the relative speeds of approach and separation.

Coefficient of Restitution ( $e$ )

The ratio of the relative speed of separation after collision to the relative speed of approach before collision.

Coefficient of Restitution

Quantifies the elasticity of a 1D collision.

e = \frac{v_{2f} - v_{1f}}{v_{1i} - v_{2i}}

Variables

Symbol	Description	Unit
$e$	Coefficient of restitution	-
$v_{1i}, v_{2i}$	Initial velocities	$\text{m/s}$
$v_{1f}, v_{2f}$	Final velocities	$\text{m/s}$

Coefficient of Restitution ( $e$ ) Concepts

$e = 1$ $e = 1$ : Perfectly elastic collision.
- $0 < e < 1$ : Inelastic collision.
- $e = 0$ : Perfectly inelastic collision (objects stick together, $v_{1f} = v_{2f}$ ).

Collisions in Two Dimensions

Vector Conservation

In two-dimensional collisions, such as a glancing blow in billiards or a vehicle intersection crash, momentum is conserved independently in both the $x$ and $y$ directions. The principles are the same as 1D collisions, but require the use of vector components.

2D Momentum Conservation Equations

The separated x and y component equations for conservation of linear momentum.

\begin{aligned} \sum p_{xi} &= \sum p_{xf} \implies m_1 v_{1ix} + m_2 v_{2ix} = m_1 v_{1fx} + m_2 v_{2fx} \\ \sum p_{yi} &= \sum p_{yf} \implies m_1 v_{1iy} + m_2 v_{2iy} = m_1 v_{1fy} + m_2 v_{2fy} \end{aligned}

Variables

Symbol	Description	Unit
$p_{x}$	Momentum in x-direction	$\text{kg} \cdot \text{m/s}$
$p_{y}$	Momentum in y-direction	$\text{kg} \cdot \text{m/s}$

Key Takeaways

Momentum ( $\vec{p} = m\vec{v}$ ) is a vector describing the "amount of motion." The net force equals the rate of change of momentum.
Impulse ( $\vec{J} = \int \vec{F} dt$ ) is a vector describing the "change in motion." It equals the change in momentum ( $\vec{J} = \Delta\vec{p}$ ).
The Conservation of Momentum principle states that the total momentum of a system is constant if no net external forces act on it.
In Elastic collisions, both momentum and kinetic energy are conserved. In Inelastic collisions, only momentum is conserved; kinetic energy is lost. The Coefficient of Restitution quantifies this.

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Learning Objectives

Center of Mass Frame

Center of Mass Frame Concepts

Center of Mass Velocity

Center of Mass Frame Concepts

Linear Momentum (ppp)

Linear Momentum (ppp) Concepts

Linear Momentum (p⃗\vec{p}p​)

Linear Momentum

Linear Momentum (ppp) Concepts

Newton's Second Law in terms of Momentum

Linear Momentum (ppp) Concepts

Impulse (JJJ)

Impulse (JJJ) Concepts

Impulse (J⃗\vec{J}J)

Impulse

Impulse (JJJ) Concepts

Impulse (JJJ) Concepts

Interactive Simulation: Impulse and Momentum

Impulse-Momentum Collision Pulse Simulator

The Impulse-Momentum Theorem

The Impulse-Momentum Theorem Concepts

The Impulse-Momentum Theorem

The Impulse-Momentum Theorem

The Impulse-Momentum Theorem Concepts

Conservation of Linear Momentum

Conservation of Linear Momentum Concepts

Conservation of Linear Momentum

Conservation of Linear Momentum

Vector Conservation of Momentum

Interactive Simulation: Collision Momentum

Momentum & Elastic Collision Simulator

Variable Mass Systems: Rocket Equation

Variable Mass Systems: Rocket Equation Concepts

Tsiolkovsky Rocket Equation

Variable Mass Systems: Rocket Equation Concepts

Types of Collisions

Types of Collisions Concepts

Coefficient of Restitution (eee)

Coefficient of Restitution (eee) Concepts

Coefficient of Restitution (eee)

Coefficient of Restitution

Coefficient of Restitution (eee) Concepts

Collisions in Two Dimensions

Vector Conservation

2D Momentum Conservation Equations

Linear Momentum ( $p$ )

Linear Momentum ( $p$ ) Concepts

Linear Momentum ( $\vec{p}$ )

Linear Momentum ( $p$ ) Concepts

Linear Momentum ( $p$ ) Concepts

Impulse ( $J$ )

Impulse ( $J$ ) Concepts

Impulse ( $\vec{J}$ )

Impulse ( $J$ ) Concepts

Impulse ( $J$ ) Concepts

Coefficient of Restitution ( $e$ )

Coefficient of Restitution ( $e$ ) Concepts

Coefficient of Restitution ( $e$ )

Coefficient of Restitution ( $e$ ) Concepts