Kinetics of Particles: Impulse and Momentum

Kinetics of Particles: Impulse and Momentum

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).

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 Equation

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

Where:

$m\mathbf{v}_1$ is the Initial Linear Momentum vector of the particle at time $t_1$ .
$\sum \int \mathbf{F} dt$ is the Linear Impulse vector of all external forces acting on the particle from $t_1$ to $t_2$ .
$m\mathbf{v}_2$ is the Final Linear Momentum vector of the particle at time $t_2$ .

Key Concepts:

Momentum ( $\mathbf{L} = m\mathbf{v}$ ): A vector quantity that characterizes the motion of a mass. Its direction is the same as the velocity.
Impulse ( $\mathbf{I} = \int \mathbf{F} dt$ ): A vector quantity representing the effect of a force acting over a period of time. For a constant force, $\mathbf{I} = \mathbf{F}(t_2 - t_1)$ .

Important

The impulse-momentum equation is a vector equation, meaning it can be resolved into independent scalar components (e.g.,

x

y

z

directions).

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

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}

Integrating the moment equation

\sum \mathbf{M}_O = \dot{\mathbf{H}}_O

with respect to time yields:

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

Where:

$(\mathbf{H}_O)_1$ is the initial angular momentum vector at $t_1$ .
$\sum \int \mathbf{M}_O \, dt$ is the sum of angular impulses of external moments about point $O$ .
$(\mathbf{H}_O)_2$ is the final angular momentum vector at $t_2$ .

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.

Motion of the Mass Center

The total linear momentum of the system can be expressed in terms of the total mass

m

and the velocity of its mass center

\mathbf{v}_G

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

Applying the principle to the mass center:

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

This shows 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.

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 Equation

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

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.

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

Collision Simulator (Impulse & Momentum)

APPROACHING

Object A

Mass (kg)

Vel (m/s)

Object B

Mass (kg)

Vel (m/s)

Restitution (

e

)0.8

Plastic (0)Elastic (1)

Initial Momentum:2.00

Final Momentum:2.00

Initial KE:22.00 J

Final KE:14.22 J

Energy Loss:7.78 J

Impact

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

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)

The coefficient of restitution,

e

, is a measure of the capacity of the colliding bodies to recover their shape after deformation. It is defined as the ratio of the relative velocity of separation to the relative velocity of approach along the line of impact.

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

Where

v_A

and

v_B

are the velocities of bodies A and B along the line of impact.

Perfectly Elastic Impact ( $e = 1$ ): No energy is lost during the collision. The objects bounce off each other perfectly.
Perfectly Plastic (Inelastic) Impact ( $e = 0$ ): The maximum amount of energy is lost. The objects stick together and move with a common final velocity.
Real Impacts ( $0 \lt e \lt 1$ ): Some kinetic energy is lost to heat, sound, or permanent deformation.

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

Coefficient of Restitution ( $e$ )

Drop a ball and see how $e$ affects the rebound height ( $h_1 = e^2 h_0$ ).

Coefficient

e

0.80

Plastic (0)Elastic (1)

Key Takeaways

Impulse and Momentum ( $m\mathbf{v}_1 + \int \mathbf{F} dt = m\mathbf{v}_2$ ) 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).