Conservative Principle and Linear Momentum

Definition:
- Conservation principles state that certain physical quantities are conserved (remain unchanged) over time in closed or isolated systems, where no external influences act.
Conserved Quantities:
- Includes:
- Conservation of energy
- Conservation of momentum
- Conservation of angular momentum
- Conservation of mass (in non-relativistic contexts)
Significance:
- Conservation laws simplify problem-solving by providing invariants.

Definition:
- A force is conservative if the work done by it on an object moving between two points is independent of the path taken and depends only on the initial and final positions.
Mathematical Formulation:
- For a force $F$ , the work $W$ along a path from $A$ to $B$ is given by:
  $W = rac{dW}{ds} <br>ightarrow ext{integrated from A to B}$
- If $W$ is path-independent, then $F$ is a conservative force.
- Equivalently:
- The curl of $F$ is zero:
  $<br>abla imes F = 0$
- $F$ can be expressed as the negative gradient of a scalar potential energy function $V$ :
  $F = -<br>abla V$

Gravitational Force:
- Equation: $F = -mg extbf{j}$ (considering near Earth's surface)
- Potential Energy: $V = mgh$
Electrostatic Force (Coulomb's Law):
- The work done is path-independent.
Spring Force (Hooke's Law):
- Equation: $F = -kx extbf{i}$
- Potential Energy: $V = rac{1}{2}kx^2$

Definition:
- Forces where work depends on the path taken, such as friction or drag.
Work in Non-Conservative Forces:
- For friction, work dissipates energy as heat, violating path independence.

Mechanics of Conservative Forces:
- The total mechanical energy $E = K + V$ (kinetic + potential) is conserved in an isolated system:
  $riangle K + riangle V = 0 <br>ightarrow K_f + V_f = K_i + V_i$
Proof:
- From the Work-Energy Theorem: $W = riangle K$ .
- For conservative forces: $W = - riangle V$ , so $riangle K = - riangle V$ .

Pendulum Motion:
- Energy oscillates between kinetic and potential.
Roller Coasters:
- Design loops using energy conservation to ensure safe speeds.

Definition:
- Linear momentum $p$ of a particle is defined as:
  $p = mv$
- For a system of particles, the total momentum $P$ is:
  $P = extstyle{\bigoplus} p_i$
Conservation Statement:
- The conservation of linear momentum states that in any system of mutually interacting or impinging particles, the linear momentum in any fixed direction remains constant unless external forces act on it in that direction.
- In an isolated system (with no external forces), the total momentum remains constant:
  $ext{Total momentum} = P_{initial} = P_{final}$
Connection to Newton's Third Law:
- Internal forces cancel in pairs, thus the net change in momentum is zero.

Momentum Conservation:
- Momentum conservation applies to all collisions, but kinetic energy may not be conserved.
Elastic Collision:
- Both momentum and kinetic energy are conserved.
- For 1D head-on collisions:
- Masses involved: $m_1, m_2$
- Momentum before and after:
  $m_1 v_{1_i} + m_2 v_{2_i} = m_1 v_{1_f} + m_2 v_{2_f}$
- Kinetic energy before and after:
  $rac{1}{2} m_1 v_{1_i}^2 + rac{1}{2} m_2 v_{2_i}^2 = rac{1}{2} m_1 v_{1_f}^2 + rac{1}{2} m_2 v_{2_f}^2$
- Solution for final velocities is provided with suitable equations.
Inelastic Collision:
- Momentum is conserved, but kinetic energy is not.
- Example: Objects stick together post-collision.
- Coefficient of restitution:
  e < 1
- Perfectly Inelastic Collision:
- Final velocity calculation:
  $v_f = rac{m_1 v_{1_i} + m_2 v_{2_i}}{m_1 + m_2}$
- Examples of Applications:
- Rocket propulsion: Conservation of momentum in exhaust gas and rocket.
- Recoil of a gun: Bullet and gun momenta equal and opposite.
- Billiard balls: Elastic collisions approximate momentum conservation.

Approximate Conservation:
- If external forces are present but impulsive (short duration), momentum is approximately conserved (e.g., explosions).