Kinetics of Particles: Impulse and Momentum - Theory & Concepts

Kinetics of Particles: Impulse and Momentum

$Learning Objectives$

Understand the principles of linear impulse and momentum.
Apply the impulse-momentum equation to single particles and systems of particles.
Analyze conservation of linear and angular momentum.
Solve impact and collision problems using the coefficient of restitution.

The principle of impulse and momentum is an alternative method to Newton's Second Law for solving kinetics problems. It is particularly useful when the problem involves forces acting over a specific interval of time, or when dealing with impulsive forces (very large forces acting over a very short time, such as impacts).

$Linear Momentum$

A vector quantity ( $\mathbf{L} = m\mathbf{v}$ ) that characterizes the motion of a mass. Its direction is the same as the velocity.

$Linear Impulse$

A vector quantity ( $\mathbf{I} = \int \mathbf{F} dt$ ) representing the effect of a force acting over a period of time. For a constant force, $\mathbf{I} = \mathbf{F}(t_2 - t_1)$ .

Principle of Linear Impulse and Momentum

Integrating Newton's Second Law ( $\sum \mathbf{F} = m \frac{d\mathbf{v}}{dt}$ ) with respect to time yields the principle of linear impulse and momentum.

$Linear Impulse and Momentum Principle$

The principle states that the initial momentum of a particle at time $t_1$ plus the sum of all impulses applied from $t_1$ to $t_2$ equals the final momentum of the particle at time $t_2$ .

Linear Impulse and Momentum Equation

Relates initial momentum, applied linear impulse, and final momentum of a particle.

m\mathbf{v}_1 + \sum \int_{t_1}^{t_2} \mathbf{F} dt = m\mathbf{v}_2

Variables

Symbol	Description	Unit
$m\mathbf{v}_1$	Initial linear momentum vector of the particle at time t_1	N·s
$\sum \int \mathbf{F} dt$	Linear impulse vector of all external forces acting from t_1 to t_2	N·s
$m\mathbf{v}_2$	Final linear momentum vector of the particle at time t_2	N·s

$Vector Resolution of Impulse-Momentum$

The impulse-momentum equation is a vector equation, meaning it can be resolved into independent scalar components (e.g., $x$ , $y$ , $z$ directions).

$Angular Momentum$

The angular momentum $\mathbf{H}_O$ of a particle about point $O$ is defined as the moment of its linear momentum: $\mathbf{H}_O = \mathbf{r} \times m\mathbf{v}$ .

Angular Impulse and Momentum for Particles

In addition to linear impulse and momentum, the principle can be extended to moments about a fixed point $O$ . This is specifically critical for analyzing central force motion where the moment of the central force about the origin is identically zero.

Angular Impulse and Momentum Equation

Relates initial angular momentum, applied angular impulse, and final angular momentum about point O.

(\mathbf{H}_O)_1 + \sum \int_{t_1}^{t_2} \mathbf{M}_O \, dt = (\mathbf{H}_O)_2

Variables

Symbol	Description	Unit
$(\mathbf{H}_O)_1$	Initial angular momentum vector at t_1	N·m·s
$\sum \int \mathbf{M}_O \, dt$	Sum of angular impulses of external moments about point O	N·m·s
$(\mathbf{H}_O)_2$	Final angular momentum vector at t_2	N·m·s

$Conservation of Angular Momentum$

If the resultant moment about point $O$ is zero (such as in orbital motion under the influence of gravity alone), the angular momentum about $O$ is conserved:

(\mathbf{H}_O)_1 = (\mathbf{H}_O)_2

This directly implies $r_1 m v_{\theta1} = r_2 m v_{\theta2}$ , which is Kepler's Second Law.

System of Particles

The principle of impulse and momentum can be extended to a system of particles. For a system of $n$ particles, the internal forces (forces acting between particles) occur in equal and opposite collinear pairs due to Newton's Third Law. The sum of their impulses over any time interval is zero. Therefore, only external forces change the total momentum of the system.

Total Linear Momentum of a System

Expresses the total linear momentum in terms of total mass and the velocity of its mass center.

\sum m_i \mathbf{v}_i = m\mathbf{v}_G

Variables

Symbol	Description	Unit
$m_i$	Mass of an individual particle	kg
$\mathbf{v}_i$	Velocity of an individual particle	m/s
$m$	Total mass of the system	kg
$\mathbf{v}_G$	Velocity of the mass center	m/s

$Motion of the Mass Center$

Applying the impulse-momentum principle to the mass center demonstrates that a system of particles moves such that its mass center acts as if all mass is concentrated there, and all external forces act on it. This is a foundational bridge to Rigid Body Kinetics.

Impulse and Momentum of the Mass Center

Applies the impulse and momentum principle to the center of mass of a system.

m(\mathbf{v}_G)_1 + \sum \int_{t_1}^{t_2} \mathbf{F}_{ext} dt = m(\mathbf{v}_G)_2

Variables

Symbol	Description	Unit
$m(\mathbf{v}_G)_1$	Initial momentum of the mass center	N·s
$\sum \int \mathbf{F}_{ext} dt$	Sum of external impulses	N·s
$m(\mathbf{v}_G)_2$	Final momentum of the mass center	N·s

$Solving Impulse and Momentum Problems$

STEP-BY-STEP

Identify the System: Determine if the problem involves a single particle or a system of interacting particles.
Draw Momentum and Impulse Diagrams: Sketch the initial momentum vectors, the impulse vectors of all external forces, and the final momentum vectors.
Establish a Coordinate System: Choose axes to resolve the vector impulse-momentum equation into scalar components.
Apply Equations: Use $m\mathbf{v}_1 + \sum \int \mathbf{F} dt = m\mathbf{v}_2$ for each component direction.
Check for Conservation: If external impulses in any direction are zero or negligible, apply the conservation of linear momentum equation for that direction.

Conservation of Linear Momentum

If the sum of the external impulses acting on a system of particles is zero, the total linear momentum of the system is conserved (remains constant).

$Conservation Applications$

This principle is most frequently applied to problems involving impacts or collisions between two or more bodies, where the impulsive forces of interaction are internal to the system and external impulses (like gravity during a very short impact time) are negligible.

Conservation of Linear Momentum Equation

States that the initial sum of momenta equals the final sum of momenta when no external impulses act.

\sum m_i (\mathbf{v}_i)_1 = \sum m_i (\mathbf{v}_i)_2

Variables

Symbol	Description	Unit
$m_i$	Mass of particle i	kg
$(\mathbf{v}_i)_1$	Initial velocity of particle i	m/s
$(\mathbf{v}_i)_2$	Final velocity of particle i	m/s

Impact System Boundary: One Body or the Pair

Choose the system before deciding which contact impulses belong in the momentum balance.

The cancellation applies only when both interacting bodies are inside the chosen system.

One-body system

The other body's contact impulse is external
Include it in this body's impulse–momentum equation

Interacting-pair system

The pair contact impulses are internal and equal/opposite
They cancel in total system momentum; external impulses remain

$Interactive Simulation$

Interact with the simulation below to explore collision and conservation of momentum.

Collision Impulse & Momentum Simulator

State: approaching

Collision Settings

Object A (Blue)

Mass (

m_1

)3.0 kg

Initial Velocity (

v_1

)3.0 m/s

Object B (Red)

Mass (

m_2

)2.0 kg

Initial Velocity (

v_2

)-2.0 m/s

Restitution (

e

)0.70

Step-by-Step Solver

1. Momentum Conservation

m_1 v_1 + m_2 v_2 = m_1 v_1' + m_2 v_2'

3 × 3.0 + 2 × -2.0 = 5.00 N·s

2. Restitution Relation

e(v_1 - v_2) = v_2' - v_1'

0.70 × (3.0 - (-2.0)) = 3.50 m/s

3. Solved Final Velocities

v_1' = \frac{m_1 v_1 + m_2 v_2 - m_2 e(v_1 - v_2)}{m_1 + m_2}

v_1' = -0.40 m/s

v_2' = v_1' + e(v_1 - v_2)

v_2' = 3.10 m/s

Momentum Balance (Ns)

Initial Momentum:5.00

Final Momentum:5.00

Conservation verified: 100% constant

Energy Balance (Joules)

Initial KE:17.50 J

Final KE:9.85 J

Thermal Loss:7.65 J

$Impact$

Impact occurs when two bodies collide over a very short time interval, generating relatively large internal forces.

$Impulsive vs. Non-Impulsive Forces$

During an impact (a very short time interval $\Delta t$ ), the extremely large forces generated between the colliding bodies are called impulsive forces. Their impulse $\int \mathbf{F} dt$ is significant.

Conversely, finite forces like weight (gravity) or small spring forces are non-impulsive forces. Their impulse over a sufficiently short impact interval is often small compared with the contact impulse, but neglect them only when that approximation is justified for the component and equation being analyzed.

Impact

$Types of Impact$

Line of Impact: The common normal to the surfaces in contact during the collision.
Central Impact: The mass centers of both colliding bodies lie on the line of impact.
Oblique Impact: One or both mass centers do not lie on the line of impact, or the initial velocities are not directed along the line of impact.

$Coefficient of Restitution (e)$

A dimensionless ratio of relative speed of separation to relative speed of approach, both measured along the line of impact. Under the usual passive-impact convention, $e \ge 0$ ; $e$ describes the normal relative-velocity response and does not by itself determine tangential motion or shape recovery.

Coefficient of Restitution Equation

Calculates the coefficient of restitution from velocities along the line of impact.

e = \frac{(v_B)_2 - (v_A)_2}{(v_A)_1 - (v_B)_1}

Variables

Symbol	Description	Unit
$e$	Coefficient of restitution	unitless
$(v_A)_1, (v_B)_1$	Initial velocities of bodies A and B along the line of impact	m/s
$(v_A)_2, (v_B)_2$	Final velocities of bodies A and B along the line of impact	m/s

Impact Directions: Normal and Tangential

Separate the line-of-impact response from the perpendicular contact response before resolving an oblique impact.

The coefficient of restitution does not by itself determine tangential velocity changes.

Normal: along line of impact

Restitution relates relative approach and separation speeds
Use the signed normal velocity components

Tangential: perpendicular to line of impact

Tangential velocity can change if tangential impulse acts
Neglect tangential impulse only when the contact model supports it

Particle Impact: System, Impulse, and Restitution

Choose the impact system and account for external impulse before applying the contact law.

Set a signed axis along the line of impact. The restitution relation applies to relative normal approach and separation speeds; it does not determine tangential velocity changes. Internal contact impulses cancel only when the interacting bodies are both inside the selected system. Neglect an external impulse only when that approximation is justified over the stated impact interval.

Start impact analysis → Set system, impact interval, line of impact, and signed axes; Set system, impact interval, line of impact, and signed axes → Are both interacting bodies in the system?; Are both interacting bodies in the system? — Yes → Internal pair contact impulses cancel; Are both interacting bodies in the system? — No → Include the contact impulse in this body's balance; Internal pair contact impulses cancel → Is external impulse negligible in the analyzed component?; Include the contact impulse in this body's balance → Is external impulse negligible in the analyzed component?; Is external impulse negligible in the analyzed component? — Yes → Conserve total momentum in this component; Is external impulse negligible in the analyzed component? — No → Apply impulse–momentum with external impulse; Conserve total momentum in this component → Is coefficient of restitution e specified or defined?; Apply impulse–momentum with external impulse → Is coefficient of restitution e specified or defined?; Is coefficient of restitution e specified or defined? — Yes → Apply e to relative normal approach and separation speeds; Is coefficient of restitution e specified or defined? — No → Is another normal contact law specified?; Apply e to relative normal approach and separation speeds → Include tangential impulse per contact model; Is another normal contact law specified? — Yes → Apply the specified normal contact law; Is another normal contact law specified? — No → Obtain e or another contact law; do not assume a value; Apply the specified normal contact law → Include tangential impulse per contact model; Include tangential impulse per contact model → Solve momentum and contact equations component by component; Solve momentum and contact equations component by component → Do impulse balance and contact direction agree?; Do impulse balance and contact direction agree? — Yes → Impact solution consistent; Do impulse balance and contact direction agree? — No → Recheck system, axes, impulses, and contact assumptions; Recheck system, axes, impulses, and contact assumptions → Set system, impact interval, line of impact, and signed axes; Obtain e or another contact law; do not assume a value → Stop until contact data or a law is available

Start impact analysis: terminator
Set system, impact interval, line of impact, and signed axes: process
Are both interacting bodies in the system?: decision
Internal pair contact impulses cancel: process
Include the contact impulse in this body's balance: process
Is external impulse negligible in the analyzed component?: decision
Conserve total momentum in this component: process
Apply impulse–momentum with external impulse: process
Is coefficient of restitution e specified or defined?: decision
Apply e to relative normal approach and separation speeds: process
Is another normal contact law specified?: decision
Apply the specified normal contact law: process
Obtain e or another contact law; do not assume a value: process
Include tangential impulse per contact model: process
Solve momentum and contact equations component by component: process
Do impulse balance and contact direction agree?: decision
Impact solution consistent: terminator
Recheck system, axes, impulses, and contact assumptions: process
Stop until contact data or a law is available: terminator

$Elasticity in Impact$

Perfectly Elastic Normal Impact ( $e = 1$ ): Relative speed along the line of impact is unchanged. For an isolated smooth impact, kinetic energy is conserved.
Perfectly Inelastic Normal Impact ( $e = 0$ ): Relative separation speed along the line of impact is zero. Tangential relative motion may remain; the bodies do not necessarily stick in every direction.
Passive Impacts ( $0 \lt e \lt 1$ ): Relative normal kinetic energy decreases; the amount depends on the system and any external impulse.

$Interactive Simulation$

Interact with the simulation below to explore the coefficient of restitution.

Coefficient of Restitution Simulator

e: 0.75 | h_0: 4.0 m

Simulation Settings

Restitution Coeff (

e

)0.75

Release Height (h0)4.0 m

Mathematical Derivation

Coefficient of Restitution

e = -\frac{v^+}{v^-}

Waiting for first bounce...

Peak Height Decay

h_{next} = e^2 h_{prev}

Initial Height:

h_0 = 4.00 \text{ m}

Current Bounces: 0

Next Max Height:

h_{1} = 4.00 \text{ m}

Current Time0.00 s

Height (y)4.00 m

Velocity (v)0.00 m/s

Key Takeaways

Impulse and Momentum Principle relates forces, mass, velocities, and time directly.
Linear Momentum ( $m\mathbf{v}$ ) is the "quantity of motion."
Linear Impulse ( $\int \mathbf{F} dt$ ) is the effect of a force over a time interval.
System of Particles: The mass center acts as a single particle with mass $m$ experiencing the net external impulse.
Conservation of Momentum occurs when net external impulses are zero, commonly used in collision problems.
Coefficient of Restitution ( $e$ ) defines the elasticity of an impact, ranging from 0 (plastic) to 1 (elastic).

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