Momentum, Impulse, Work, Energy, and Collisions: Study Notes

Principle of Linear Impulse and Momentum

Linear Momentum

Linear momentum is a fundamental concept in physics that describes the quantity of motion an object possesses. It is defined as the product of an object's mass and its velocity.

• Definition: The linear momentum p of a particle is given by $p = m v$.

• Conservation: In the absence of external forces, the total linear momentum of a system remains constant.

• Vector Quantity: Momentum has both magnitude and direction.

Impulse

Impulse is the effect of a force acting over a period of time, resulting in a change in momentum.

• Definition: Impulse I is given by $I = F riangle t$ or more generally $I = \int F dt$.

• Relation to Momentum: The impulse delivered to an object equals the change in its momentum: $I = \triangle p$.

Principle of Linear Impulse and Momentum

• Equation: $\sum F dt = m v_2 - m v_1$

• System of Particles: $\sum I_{initial} + \int \sum F_{ext} dt = \sum I_{final}$

• Application: Used to analyze collisions, explosions, and other interactions.

Conservation of Linear Momentum in Collisions

Conservation Principle

During a collision, if no external force acts on the system, the total momentum before the collision equals the total momentum after.

• Equation: $m_A v_{A1} + m_B v_{B1} = m_A v_{A2} + m_B v_{B2}$

• Example: Two objects colliding and moving apart; their combined momentum remains constant.

Types of Collisions

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not.

• Perfectly Inelastic Collision: The colliding objects stick together after impact.

Coefficient of Restitution

Definition and Formula

The coefficient of restitution (e) quantifies the elasticity of a collision, representing the ratio of relative speeds after and before impact.

• Equation: $e = \frac{(v_{B2} - v_{A2})}{(v_{A1} - v_{B1})}$

• Range: $0 \leq e \leq 1$

• Interpretation: e = 1 for perfectly elastic collisions; e = 0 for perfectly inelastic collisions.

Application in Collisions

• Used to determine post-collision velocities.

• Depends on material properties and shape.

Impact: Central and Oblique

Central Impact

Occurs when the line of impact passes through the centers of mass of the colliding bodies.

• Analysis: Only the velocities along the line of impact are considered.

• Equation: $m_A v_{A1} + m_B v_{B1} = m_A v_{A2} + m_B v_{B2}$

Oblique Impact

Occurs when the line of impact does not pass through the centers of mass, and velocities must be resolved into components.

• Analysis: Velocities are split into components parallel and perpendicular to the line of impact.

• Equation: $e = \frac{(v_{B2})_x - (v_{A2})_x}{(v_{A1})_x - (v_{B1})_x}$

Principle of Work and Energy

Work-Energy Principle

The work done by all forces acting on a particle equals the change in its kinetic energy.

• Equation: $T_1 + \sum U_{1-2} = T_2$

• Kinetic Energy: $T = \frac{1}{2} m v^2$

• Potential Energy: $U = mgh$ (gravitational), $U = \frac{1}{2} k x^2$ (spring)

Conservation of Energy

If only conservative forces do work, the total mechanical energy (kinetic + potential) of a system remains constant.

• Equation: $T_1 + V_1 = T_2 + V_2$

• Application: Used in analyzing motion under gravity, springs, and other conservative forces.

Conservative Forces and Potential Energy

Conservative Forces

A force is conservative if the work it does depends only on the initial and final positions, not the path taken.

• Examples: Gravitational force, spring force

• Equation: $U = -\int F dx$

Potential Energy

• Gravitational: $U = mgh$

• Spring: $U = \frac{1}{2} k x^2$

Power and Efficiency

Power

Power is the rate at which work is done or energy is transferred.

• Equation: $P = \frac{dU}{dt} = F \cdot v$

• Units: 1 Watt (W) = 1 Joule/second (J/s)

Efficiency

Efficiency is the ratio of useful power output to total power input.

• Equation: $E = \frac{\text{Power output}}{\text{Power input}}$

• Note: Efficiency is always less than 1.

Example Problems

Collision Example

• Two balls collide; use conservation of momentum and coefficient of restitution to find final velocities.

• Given: $m_A$, $m_B$, $v_{A1}$, $v_{B1}$, $e$

• Find: $v_{A2}$, $v_{B2}$ using equations above.

Impulse Example

• A person dives from a boat; use conservation of momentum to find the velocity of the boat after the dive.

• Given: $m_{person}$, $m_{boat}$, $v_{person}$

• Find: $v_{boat}$

Work and Energy Example

• A block slides down a frictionless incline; use conservation of energy to find its velocity at the bottom.

• Given: $h$, $m$

• Find: $v$ using $mgh = \frac{1}{2} m v^2$

Summary Table: Key Equations

Concept	Equation (LaTeX)	Description
Linear Momentum	$p = m v$	Product of mass and velocity
Impulse	$I = \int F dt$	Force over time interval
Conservation of Momentum	$m_A v_{A1} + m_B v_{B1} = m_A v_{A2} + m_B v_{B2}$	Total momentum before = after
Coefficient of Restitution	$e = \frac{(v_{B2} - v_{A2})}{(v_{A1} - v_{B1})}$	Elasticity of collision
Kinetic Energy	$T = \frac{1}{2} m v^2$	Energy due to motion
Potential Energy (Gravity)	$U = mgh$	Energy due to position
Potential Energy (Spring)	$U = \frac{1}{2} k x^2$	Energy stored in spring
Power	$P = F \cdot v$	Rate of doing work
Efficiency	$E = \frac{\text{Power output}}{\text{Power input}}$	Ratio of useful power

Additional info: Some equations and examples have been expanded for clarity and completeness. Diagrams referenced in the notes illustrate collision types, impulse, and energy conservation scenarios.