Force and Motion - Momentum, Impulse, Collisions, and Rocket Propulsion

Flashcards testing definitions, formulas, quantitative specifics, and key concepts of force, momentum, impulse, elastic and inelastic collisions in 1D and 2D, and rocket propulsion.

Linear Momentum

The product of mass and velocity of a moving body, given by the formula $\vec{p} = m\text{v}$, which describes the quality and quantity of motion.

SI Unit and Dimensions of Momentum

The SI unit of momentum is $\text{N}\cdot\text{s}$ or $\text{kg}\,\text{m}\,\text{s}^{-1}$, and its dimensional formula is $[\text{MLT}^{-1}]$.

Newton's Second Law of Motion (in terms of Momentum)

The principle stating that the time rate of change of momentum of a body is equal to the applied force, written mathematically as $\vec{F} = \frac{\Delta \vec{p}}{\Delta t}$.

Impulse

The product of average force and the time interval for which it acts ($\vec{I} = \vec{F} \times t = \Delta \vec{p}$), representing the instantaneous change in momentum of a body.

Impulsive Force

A force that acts on a body for a very short interval of time, varying from instant to instant during impact.

Isolated System

A system on which no external agency exerts any force, such as gas molecules enclosed in a glass vessel at constant temperature.

Law of Conservation of Linear Momentum

The rule stating that the total linear momentum of an isolated system remains constant ($m_1\vec{v}_1 + m_2\vec{v}_2 = m_1\vec{v}'_1 + m_2\vec{v}'_2$).

Elastic Collision

An ideal collision in which the kinetic energy of the system as well as total momentum and total energy are conserved ($\text{K.E.}_i = \text{K.E.}_f$).

Inelastic Collision

A real-world collision in which kinetic energy is not conserved ($\text{K.E.}_i > \text{K.E.}_f$) because energy is lost as sound, heat, or deformation, though total momentum remains conserved.

Relative Velocity Law of 1D Elastic Collision

The condition showing that the magnitude of relative velocity of approach equals the magnitude of relative velocity of separation, but reversed after collision ($v_1 - v_2 = -(v'_1 - v'_2)$).

1D Elastic Collision: Equal Masses ($m_1 = m_2$)

When two bodies of equal mass collide elastically in one dimension, they interchange their velocities after the collision ($v'_1 = v_2$ and $v'_2 = v_1$).

1D Elastic Collision: Lighter Mass Colliding with Massive Target at Rest ($m_1 \ll m_2$, $v_2 = 0$)

The case where the lighter body $m_1$ bounces back with its initial speed ($v'_1 = -v_1$) while the massive target $m_2$ remains at rest ($v'_2 = 0$).

1D Elastic Collision: Massive Mass Colliding with Lighter Target at Rest ($m_1 \gg m_2$, $v_2 = 0$)

The case where the massive body $m_1$ continues with its original velocity ($v'_1 = v_1$) while the lighter target $m_2$ moves forward with double the initial speed ($v'_2 = 2v_1$).

Coefficient of Restitution (Perfectly Elastic Collision)

A scalar measure whose numerical value is $1$ for a perfectly elastic collision.

Perfectly Inelastic Collision in 1D

A collision where the colliding bodies stick together after impact and move with a common final velocity ($v_f = \frac{m_1 v_1 + m_2 v_2}{m_1 + m_2}$).

2D Elastic Collision Momentum Equations

Components of linear momentum conservation resolved along x-axis ($m_1 v_1 + m_2 v_2 = m_1 v'_1 \cos(\theta_1) + m_2 v'_2 \cos(\theta_2)$) and y-axis ($0 = m_1 v'_1 \sin(\theta_1) - m_2 v'_2 \sin(\theta_2)$).

Rocket Propulsion

The acceleration mechanism where ignited fuel turns to high-pressure gas expelled backward at high velocity, gaining forward momentum equal to the expelled gas momentum.

Rocket Acceleration Formula

The mathematical equation $a = \frac{m v}{M}$, where $m$ is mass of gas ejected per second, $v$ is ejection velocity relative to rocket, and $M$ is total instantaneous rocket mass.

Rocket Fuel Consumption Parameters

A typical rocket consumes about $10000\,\text{kg}\,\text{s}^{-1}$ of fuel, ejects burnt gases at speeds over $4000\,\text{m}\,\text{s}^{-1}$, and consists of more than $80\%$ launch mass as fuel.

Rocket Engine Diagram

Illustration showing liquid hydrogen fuel and liquid oxygen mixing in the combustion chamber, exhausting hot gases at high velocity to produce equal and opposite rocket momentum.

Elastic Collision in 2D Diagram

Diagram depicting two non-rotating balls $m_1$ and $m_2$ colliding at origin and scattering at angles $\theta_1$ and $\theta_2$ relative to the x-axis.

Hair Crumple Zone Thresholds

The physiological property where skin and hair extend impact time, raising the force required to fracture the cranium from $5\,\text{N}$ (naked skull) to $50\,\text{N}$.