Chapter 8: Linear Momentum and Collisions – Study Notes

Linear Momentum and Collisions

Introduction

In many physical situations, such as a bullet hitting a carrot or hailstones shattering a roof, the forces involved are too complex to analyze using Newton's second law alone. Instead, the concepts of momentum and impulse, along with the conservation of momentum, provide powerful tools for solving these problems.

Units of Chapter 8

• Linear Momentum

• Momentum and Newton's Second Law

• Impulse

• Conservation of Linear Momentum

• Inelastic Collisions

• Elastic Collisions

This chapter introduces momentum as a new conserved quantity, distinct from energy, and explores its applications in various types of collisions.

Linear Momentum

Definition and Properties

Linear momentum (p) is a vector quantity defined for a single particle as:

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

• Direction: Same as the velocity vector.

• SI Units: kg·m/s

• Component Form: $p_x = m v_x$, $p_y = m v_y$, $p_z = m v_z$

Momentum distinguishes between objects of different mass and velocity. For example, a bowling ball moving at the same speed as a tennis ball has much greater momentum due to its larger mass.

Factors Affecting Momentum

• Mass: More mass means more momentum for the same velocity.

• Velocity: Higher velocity increases momentum, even for small masses.

Examples:

• A bus moving slowly can have large momentum due to its mass.

• A bullet moving fast can have large momentum despite its small mass.

Momentum as a Vector

• Momentum must be added using vector addition, not by simply adding magnitudes.

• Direction matters: e.g., momentum north is different from momentum east, even at the same speed.

Example: Two objects moving at right angles require vector addition to find total momentum.

Change in Momentum

• When an object changes direction or speed, its momentum changes.

• Example: A beanbag bear comes to rest after hitting the floor (change in momentum = $mv$ upward). A rubber ball bounces upward with speed $V$ (change in momentum = $2mv$ upward).

Example Calculation

A 2.5-kg stone is released from rest and falls for 4.0 s. Its momentum is:

$p = m v = 2.5 \times (9.8 \times 4.0) = 98$ kg·m/s

Momentum and Newton's Second Law

Relation to Force

Newton's second law can be expressed in terms of momentum:

$\sum \vec{F} = m \frac{d\vec{v}}{dt}$

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

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

• If the net external force is zero, momentum is conserved.

Conservation of Linear Momentum

Principle of Conservation

If the net external force acting on a system is zero, the total momentum of the system remains constant:

$\vec{p}_f = \vec{p}_i$

• Internal forces (forces between objects within the system) always sum to zero due to Newton's third law.

• Internal forces cannot change the total momentum of the system.

Isolated Systems

• An isolated system has no net external forces acting on it.

• Example: Two astronauts pushing off each other in space.

Everyday Example

• When you push someone on ice, both move in opposite directions. The system starts and ends with zero total momentum.

Formal Statement

The law of conservation of momentum states: The total momentum of an isolated system of objects remains constant.

Example: Conservation in Practice

A child in a boat throws a package horizontally. The boat moves in the opposite direction to conserve momentum:

$p_{initial} = 0$

$p_{final} = (26 + 45)v + (6.4)(10.0)$

$v = -0.90$ m/s

Collisions

Definition and Types

A collision is a strong interaction between bodies that lasts a short time. Examples include car crashes, billiard balls, and particles in accelerators.

• If internal forces are much larger than external forces, the system can be treated as isolated.

• Momentum is conserved in collisions.

• Types: inelastic and elastic collisions.

Inelastic Collisions

In an inelastic collision, some kinetic energy is lost to other forms (thermal, potential, chemical, or nuclear energy). A perfectly inelastic collision occurs when objects stick together after the collision and move with a common velocity.

• Examples: Clay hitting the floor, bullet embedding in wood, railroad cars coupling, dart hitting a board.

• Momentum is conserved, but kinetic energy is not.

Elastic Collisions

In an elastic collision, both momentum and kinetic energy are conserved. Collisions between hard objects (e.g., billiard balls, steel balls) approximate elastic collisions.

• Most real-world collisions are not perfectly elastic; some energy is always lost.

Comparison Table: Elastic vs. Inelastic Collisions

Type	Momentum Conserved?	Kinetic Energy Conserved?	Example
Elastic	Yes	Yes	Billiard balls, steel balls
Inelastic	Yes	No	Car crashes, clay hitting floor
Perfectly Inelastic	Yes	No (maximum loss)	Objects stick together

Key Equations for Collisions

• Momentum conservation (one dimension): $m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f}$

• Perfectly inelastic collision: $v_f = \frac{m_1 v_{1i} + m_2 v_{2i}}{m_1 + m_2}$

• Elastic collision (kinetic energy conserved): $K_i = K_f$

Examples of Collisions

• Hockey puck and chest: A puck embeds in a chest on ice. Use conservation of momentum to find the puck's speed.

• Football players: Two players collide and stick together. Add momentum vectors to find final direction and speed.

• Car collision at intersection: Use vector addition and conservation of momentum to find the direction and magnitude of the wreckage's velocity.

Sample Calculation: Car Collision at Intersection

Given: 1500-kg car at 25.0 m/s east, 2500-kg truck at 20.0 m/s north.

Find direction and speed after collision (vehicles stick together):

$\theta = \tan^{-1}\left(\frac{m_t v_t}{m_c v_c}\right) = \tan^{-1}\left(\frac{2500 \times 20.0}{1500 \times 25.0}\right) = 53.1^\circ$

$v_f = \frac{\sqrt{(2500 \times 20.0)^2 + (1500 \times 25.0)^2}}{1500 + 2500} = 15.6$ m/s

Summary Table: Conservation Laws in Collisions

Collision Type	Momentum	Kinetic Energy
Elastic	Conserved	Conserved
Inelastic	Conserved	Not conserved
Perfectly Inelastic	Conserved	Maximum loss

Key Points

• Momentum is always conserved in collisions, regardless of whether kinetic energy is conserved.

• Elastic collisions conserve both momentum and kinetic energy.

• Inelastic collisions conserve momentum but not kinetic energy.

• Perfectly inelastic collisions result in objects sticking together after the collision.

Additional info:

• Impulse, which is the change in momentum due to a force acting over a time interval, is also a key concept in this chapter but was not covered in detail in the provided materials.

• Vector addition is essential for analyzing momentum in two or three dimensions.