Momentum, Impulse, and Collisions – Study Notes

Chapter 8: Momentum, Impulse, and Collisions

Learning Goals

• Understand the momentum of a particle and how net force changes momentum.

• Identify when the total momentum of a system is conserved.

• Distinguish between elastic, inelastic, and completely inelastic collisions.

• Define the center of mass of a system and describe its motion.

• Analyze systems with changing mass, such as rocket propulsion.

Introduction

In many real-world situations, such as hailstones damaging a roof, the forces involved are too complex for direct application of Newton's second law. In these cases, the concepts of momentum, impulse, and the conservation of momentum provide powerful tools for analysis.

Momentum and Newton's Second Law

Definition of Momentum

• Momentum (p) is a vector quantity defined as the product of an object's mass and its velocity.

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

• The direction of momentum is the same as the direction of velocity.

Newton's Second Law in Terms of Momentum

• Newton's second law can be written as:

  $\sum \vec{F} = \frac{d\vec{p}}{dt}$

• The net force on a particle equals the rate of change of its momentum.

Impulse

Definition of Impulse

• Impulse (J) is the product of the force and the time interval over which it acts.

• For a constant force:

  $\vec{J} = \vec{F}_{\text{net}} \Delta t$

• For a varying force, impulse is the area under the force vs. time curve:

  $\vec{J} = \int_{t_i}^{t_f} \vec{F}_{\text{net}}\,dt$

Impulse-Momentum Theorem

• The change in momentum of a particle equals the impulse of the net force acting on it:

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

• This theorem is especially useful when forces are large and act over short times (e.g., collisions).

Applications and Examples

• Landing with bent knees increases the stopping time, reducing the force on the body.

• In car crashes, crumple zones increase the time over which the car stops, reducing the force on occupants.

Momentum vs. Kinetic Energy

• Kinetic energy is related to the work done by a force over a distance:

$K = \frac{1}{2}mv^2$

• Momentum is related to the impulse imparted by a force over a time interval.

• Both are conserved in certain types of collisions, but under different conditions.

Conservation of Momentum

Isolated Systems

• An isolated system is one in which no external forces act.

• In such systems, the total momentum is conserved:

$\sum \vec{F}_{\text{ext}} = 0 \implies \vec{p}_{\text{total, initial}} = \vec{p}_{\text{total, final}}$

• Momentum is a vector; use vector addition when combining momenta.

Example: Astronauts in Space

• Two astronauts push off each other in space. The forces they exert are internal, so the total momentum of the system remains constant.

Types of Collisions

Elastic Collisions

• Both momentum and kinetic energy are conserved.

• Common in collisions between hard objects (e.g., billiard balls).

Inelastic Collisions

• Momentum is conserved, but kinetic energy is not.

• Some kinetic energy is transformed into other forms (e.g., heat, deformation).

Completely Inelastic Collisions

• Colliding objects stick together after the collision.

• Maximum possible loss of kinetic energy consistent with momentum conservation.

Summary Table: Types of Collisions

Type	Momentum Conserved?	Kinetic Energy Conserved?	Objects Stick Together?
Elastic	Yes	Yes	No
Inelastic	Yes	No	No
Completely Inelastic	Yes	No	Yes

One-Dimensional Elastic Collisions

• For two bodies A and B, with B initially at rest:

$v_{A,\text{final}} = \frac{m_A - m_B}{m_A + m_B} v_{A,\text{initial}}$

$v_{B,\text{final}} = \frac{2m_A}{m_A + m_B} v_{A,\text{initial}}$

• Special cases:

  • If $m_B \gg m_A$, A bounces back with nearly the same speed.

  • If $m_A = m_B$, A stops and B moves with A's initial speed.

  • If $m_B \ll m_A$, A slows slightly, B moves with nearly twice A's initial speed.

Center of Mass

Definition

• The center of mass of a system is the point where the system's mass can be considered to be concentrated for translational motion.

$\vec{R}_{\text{cm}} = \frac{1}{M} \sum_{i} m_i \vec{r}_i$

• For symmetric, homogeneous objects, the center of mass is at the geometric center.

• For objects with an axis of symmetry, the center of mass lies along that axis.

Motion of the Center of Mass

• The total momentum of a system equals the total mass times the velocity of the center of mass:

$\vec{p}_{\text{total}} = M \vec{v}_{\text{cm}}$

• When external forces act, the center of mass moves as if all mass were concentrated at that point and all external forces acted there.

Example: Exploding Shell

• Fragments follow individual paths, but the center of mass continues along the original trajectory (if no external forces act).

Applications and Problem Types

• Calculating impulse and average force during collisions.

• Analyzing momentum conservation in isolated systems (e.g., astronauts, skaters).

• Solving for final velocities in elastic and inelastic collisions.

• Determining the motion of the center of mass in multi-object systems.

Key Equations Summary

• Momentum: $\vec{p} = m\vec{v}$

• Impulse: $\vec{J} = \int \vec{F}_{\text{net}} dt$

• Impulse-Momentum Theorem: $\vec{J} = \Delta \vec{p}$

• Conservation of Momentum: $\vec{p}_{\text{initial}} = \vec{p}_{\text{final}}$ (if $\sum \vec{F}_{\text{ext}} = 0$)

• Center of Mass: $\vec{R}_{\text{cm}} = \frac{1}{M} \sum m_i \vec{r}_i$

Additional info: These notes are based on the introductory slides and learning goals for Chapter 8 of University Physics with Modern Physics, focusing on momentum, impulse, collisions, and center of mass. For detailed worked examples and problem-solving strategies, refer to the full textbook chapter.