module 4 lecture 7: Momentum, Collisions, and Impulse

Momentum, Collisions, and Impulse

Introduction

This module covers the fundamental concepts of momentum, types of collisions, the law of conservation of momentum, and impulse. These principles are essential for understanding the behavior of objects in motion and during interactions such as collisions.

Momentum

Definition and Properties

• Momentum is a measure of the quantity of motion an object possesses.

• It is defined as the product of an object's mass and velocity.

• Momentum is a vector quantity, meaning it has both magnitude and direction.

• Formula:

$p = m v$

• Where p is momentum (kg·m/s), m is mass (kg), and v is velocity (m/s).

• Unit of Momentum: kg·m/s

Examples

• A moving boulder has more momentum than a stone rolling at the same speed due to its greater mass.

• A fast boulder has more momentum than a slow boulder of the same size.

• An object at rest has zero momentum.

Conceptual Questions

• Which has greater momentum: a heavy truck at rest or a moving skateboard? Answer: The moving skateboard, because the truck's velocity is zero.

• If the speed of an object is doubled, its momentum also doubles.

• Example calculation: A 4 kg ball has a momentum of 12 kg·m/s. Its speed is $v = \frac{p}{m} = \frac{12}{4} = 3$ m/s.

Law of Conservation of Linear Momentum

Principle and Application

• The law of conservation of momentum states that in the absence of external forces, the total momentum of a system remains constant before and after a collision.

• Mathematical Statement:

$\text{Total momentum before collision} = \text{Total momentum after collision}$ $\sum m v_{\text{before}} = \sum m v_{\text{after}}$

• This law applies to all types of collisions (elastic and inelastic).

Examples

• When two bumper cars collide, they exchange momentum, but the total momentum of the system remains unchanged.

• If one car is much heavier, the lighter car will move more after the collision to conserve momentum.

Types of Collisions

Elastic and Inelastic Collisions

• Elastic Collision: No kinetic energy is lost; objects bounce off each other without deformation or heat generation.

• Inelastic Collision: Some kinetic energy is converted to other forms (e.g., heat, deformation). In a perfectly inelastic collision, objects stick together after the collision.

• Regardless of type, momentum is always conserved in the absence of external forces.

Example Calculations

• Inelastic Collision: A 10 kg bowling ball at rest and a 0.01 kg piece of putty moving at 100 m/s collide and stick together. Total momentum before = $0 + (0.01 \times 100) = 1$ kg·m/s. After collision, combined mass is 10.01 kg, and total momentum remains 1 kg·m/s.

• Identical Masses: A car of mass m moving at 10 m/s collides inelastically with another car of mass m at rest. Final velocity $V$ is: $m v_1 + m v_2 = (m + m)V$ $m \times 10 + m \times 0 = 2m V$ $V = 5$ m/s

• Head-on Collision: Two billiard balls of equal mass and speed move toward each other. Their combined momentum after collision is zero, as their momenta cancel.

Impulse

Definition and Relationship to Momentum

• Impulse is the change in momentum of an object when a force is applied over a time interval.

• Formula:

$\text{Impulse} = F \Delta t = \Delta p = m v_{\text{final}} - m v_{\text{initial}}$

• Impulse has the same units as momentum: kg·m/s or N·s.

• The greater the impulse applied to an object, the greater its change in momentum.

Applications and Examples

• Extending the time over which momentum is reduced (e.g., landing on a haystack instead of concrete) reduces the force of impact.

• Airbags in cars increase the time over which momentum is reduced, thereby reducing the force on passengers.

• Expert martial artists break bricks by applying a large force over a very short time interval, maximizing impulse.

Momentum in Explosions and Rockets

Conservation of Momentum in Systems

• When two objects initially at rest are pushed apart by an explosion, their momenta are equal in magnitude and opposite in direction, so total momentum remains zero.

• For rockets, the momentum of the exhaust gases moving backward is equal and opposite to the momentum of the rocket moving forward. The total momentum of the system remains zero (if starting from rest).

• After a firecracker bursts, the vector sum of the momenta of all fragments equals the original momentum of the firecracker.

Conceptual Questions and Newton's Third Law

Force, Impulse, and Change in Momentum

• When two objects interact (e.g., a bug hitting a windshield), the force, impulse, and change in momentum are equal in magnitude and opposite in direction for both objects, as per Newton's third law.

• Even if the effects (e.g., damage) are different, the physical quantities exchanged are equal and opposite.

Summary Table: Key Concepts

Concept	Definition	Formula	Unit
Momentum	Quantity of motion, product of mass and velocity	$p = m v$	kg·m/s
Impulse	Change in momentum due to force over time	$J = F \Delta t = \Delta p$	N·s or kg·m/s
Conservation of Momentum	Total momentum remains constant in absence of external forces	$\sum m v_{\text{before}} = \sum m v_{\text{after}}$	kg·m/s

Additional Info

• Momentum and impulse are foundational for understanding more advanced topics such as angular momentum, collisions in multiple dimensions, and the mechanics of systems with variable mass (e.g., rockets).

• For further practice, interactive simulations and quizzes are recommended (see instructor resources).