2025 H2 Collision Lecture Notes Tutor

Impulse - Impulse as Area Under an $F{-}t$ Graph- The change in momentum $\Delta p$ experienced by a body equals the net area under its force

-time graph between $t_1$ and $t_2$. - $\Delta p = \displaystyle \int_ {t_1}^{t_2} F\,dt$ - If the force is non-uniform, the exact area under the curve must be integrated; if an average force $F_{\text{avg}}$ is used, - $\Delta p = F_{\text{avg}}\,\Delta t$ (rectangular area having the same value as the curved area). - Definition- Impulse = product of average net force and the time interval during which it acts. - Units: $\text{N\,s} = \text{kg\,m\,s}^{-1}$ (identical to momentum units). - **Relationship to Newton

’s Second Law**- Starting from $F = \dfrac{dp}{dt}$ and integrating, one recovers the impulse

–momentum theorem above. ### Momentum (Revision) - Momentum: $\vec p = m\vec v$ (vector quantity). - Newton

’s 2nd law (in momentum form): $\displaystyle F = \frac{dp}{dt}$; direction of $F$ = direction of rate of change of momentum. ### Principle of Conservation of Momentum (PCM) - Statement

: Total momentum of a system remains constant provided no external resultant force acts on it. - System Definitions- System: chosen set of objects that may interact. - Closed (isolated) system: only internal forces (forces between the objects) are significant; external resultant force $=0$. - **Derivation (one

-dimensional two

-body sketch)**- Objects 1 and 2, masses $m_1, m_2$; initial velocities $u_1, u_2$; final velocities $v_1, v_2$. - Equal and opposite interaction forces (Newton

’s 3rd law) give equal and opposite impulses in time $\Delta t$, hence $\Delta p_1 + \Delta p_2 = 0$. - Therefore $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$ (valid regardless of directions). - Applicability- Works for collisions and disintegrations as long as external resultant force is negligible. ### Types of Interactions - Elastic / Perfectly Elastic Collisions- Momentum conserved. - Kinetic energy (KE) conserved: $\tfrac12 m_1 u_1^{2}+\tfrac12 m_2 u_2^{2}=\tfrac12 m_1 v_1^{2}+\tfrac12 m_2 v_2^{2}$. - Derived condition (relative speed form): - $u_1 - u_2 = v_2 - v_1$ (vector equation; sign matters). - In practice, perfect elasticity occurs only at atomic / molecular scales. - Inelastic Collisions- Momentum conserved. - Total KE not conserved; some KE → heat, sound, deformation, etc. - Objects normally separate after impact. - Perfectly (Completely) Inelastic Collisions- Extreme case of inelastic collisions where bodies coalesce and continue with a common velocity $v$. - Momentum equation simplifies to $m_1u_1 + m_2u_2 = (m_1+m_2)v$. - Summary Table- Elastic: momentum & KE conserved; objects separate; $u_1-u_2 = v_2-v_1$. - Inelastic: momentum conserved; KE not conserved; objects separate. - Perfectly Inelastic: momentum conserved; maximum KE loss; objects stick together. ### Relative

-Speed Criterion for Elastic Collisions (Derivation Outline) - From PCM: $m_1u_1+m_2u_2=m_1v_1+m_2v_2$. - From KE conservation: $m_1u_1^{2}+m_2u_2^{2}=m_1v_1^{2}+m_2v_2^{2}$. - Rearranging and dividing as shown in the appendix produces $u_1-u_2=v_2-v_1$. ### Disintegrations & Recoil Situations - System initially at rest may fragment; momentum of parts still sums to zero.- Examples: - Recoil of a gun and bullet: gun + bullet momentum before firing $=0$; after firing momenta are equal in magnitude, opposite in direction. - Bomb exploding mid

-air: vector sum of fragment momenta equals pre

-explosion momentum of bomb. - Nuclear decay: stationary nucleus emits particles so that vector sum of momenta remains zero; discrepancies historically led to discovery of neutrinos. - Energy note: KE often increases because internal chemical, nuclear or mass

-energy is converted into kinetic energy. ### Worked Examples (Condensed) - **Example 1: 1

-D Elastic Collision Between 1.6 kg & 2.4 kg Blocks**- Data: $m_1=1.6\,\text{kg},\;u_1=5.5\,\text{m s}^{-1}$ ; $m_2=2.4\,\text{kg},\;u_2=2.5\,\text{m s}^{-1}$; $v_2=4.9\,\text{m s}^{-1}$. - PCM gives $v_1=1.9\,\text{m s}^{-1}$. - KE before $=31.7\,\text{J}$; KE after $=31.7\,\text{J}$ → elastic. - Alternate check: $u_1-u_2=3.0\,\text{m s}^{-1},\;v_2-v_1=3.0\,\text{m s}^{-1}$ → satisfies relative

-speed criterion. - Example 2: Two Trolleys- Before: $p_X=20\,\text{kg m s}^{-1}$ (right), $p_Y=12\,\text{kg m s}^{-1}$ (left). - After: $p_X=+2\,\text{kg m s}^{-1}$. - PCM: $+20-12 = +2 + p_Y$ ⇒ $p_Y=+10\,\text{kg m s}^{-1}$ (right). - Example 3: Perfectly Inelastic Collision of Identical Particles- Two masses $m$, one initially at rest. - PCM: $mu = 2mv$ ⇒ $v = 0.5u$. - KE ratio: \dfrac{\text{KE_\text{after}}}{\text{KE_\text{before}}}=0.5. - Interpretation: 50 % of initial KE lost (heat, sound, deformation). - Example 4: Conceptual – Bungee Jumper- Individual

’s momentum changes (external force of gravity present) → no PCM violation if entire Earth

–rope

–man system considered. - **Example 5: Nuclear Disintegration $$\;^{235}$