💥 Momentum

——— ⚡ Momentum ———

~ 📘 Definitions ~

Momentum (p):
The product of mass and velocity of an object.
Formula: $ p = m \times v $
It is a vector quantity (has both magnitude and direction).
Momentum represents the “amount of motion” an object has.
Relation to force:
Force causes changes in momentum. The rate of momentum change equals the force applied:

$$F = \frac{\Delta p}{\Delta t}$$

~ 🌀 Impulse ~

Impulse (J) is the change in momentum caused by a force acting over a time interval.
Formula:
$$J = F_{\text{avg}} \times \Delta t = \Delta p = m v_f - m v_i$$
Where:

F_avg = average force applied
Δt = time interval force acts
Δp = change in momentum

Impulse is also a vector and has units of kg·m/s or N·s.
If force varies with time, impulse is the integral of force over time:

$$J = \int F(t) \, dt$$
The impulse-momentum theorem states:

$$J = \Delta p$$

——— ♻️ Momentum Conservation ———

~ 📙 Definition ~

In a closed system (no external forces), total momentum remains constant.
When objects interact (collide or push), total momentum before equals total momentum after:

$$\sum p_{\text{initial}} = \sum p_{\text{final}}$$

$$m_1 v_{1i} + m_2 v_{2i} + \cdots = m_1 v_{1f} + m_2 v_{2f} + \cdots$$

Example: Two ice skaters push off each other on frictionless ice.

Skater 1 moves forward with momentum $ p_1 $
Skater 2 moves backward with momentum $ p_2 $
Because the system started at rest, $ p_1 + p_2 = 0 $

~ 💡 Notes ~

Momentum is a vector — direction matters.
External forces (like friction) can change total momentum.
Conservation of momentum helps analyze collisions and explosions.

——— ⚔️ Elastic vs. Inelastic Collisions ———

1) Elastic Collisions

Objects bounce without losing kinetic energy.
Both momentum and kinetic energy are conserved.
Example: Two billiard balls colliding.

Formulas:

$$\text{Momentum before} = \text{Momentum after}$$

$$\text{Kinetic Energy before} = \text{Kinetic Energy after}$$

2) Inelastic Collisions

Objects lose some kinetic energy (heat, sound, deformation).
Momentum is conserved, but kinetic energy is not.
In perfectly inelastic collisions, objects stick and move together.
Example: Clay lump sticking to the floor.

Formulas:

$$\text{Momentum before} = \text{Momentum after}$$

$$\text{Kinetic Energy before} > \text{Kinetic Energy after}$$

——— ⚖️ Center of Mass ———

~ 📚 Definition ~

The point where mass of an object or system is considered concentrated.
It's the weighted average position of all mass.

For particles:

$$\vec{R}_{cm} = \frac{1}{M} \sum_i m_i \vec{r}_i$$

$ \vec{R}_{cm} $ = center of mass position vector
$ M = \sum_i m_i $ total mass
$m_i$ = mass of i^th particle
$ \vec{r}_i $ = position vector of i^th particle

For continuous objects:

$$\vec{R}_{cm} = \frac{1}{M} \int \vec{r} \, dm$$

The center of mass moves as if all external forces act on a single point mass located there.

——— 💥 Momentum ———