Collisions — Revision Notes

⚡ 30-Second Revision

Momentum: — $\vec{p} = m\vec{v}$ (vector quantity)\n- Impulse: $\vec{J} = \Delta \vec{p} = \vec{F}_{avg} \Delta t$\n- Conservation of Momentum: $m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2$ (always conserved in isolated system)\n- Elastic Collision: Momentum conserved, Kinetic Energy conserved ($e=1$)\n - 1D Equal Masses: Velocities exchanged ($v_1=u_2, v_2=u_1$)\n - 1D General: $v_1 = \frac{(m_1 - m_2)u_1 + 2m_2u_2}{m_1 + m_2}$, $v_2 = \frac{(m_2 - m_1)u_2 + 2m_1u_1}{m_1 + m_2}$\n- Inelastic Collision: Momentum conserved, Kinetic Energy NOT conserved ($0 < e < 1$)\n- Perfectly Inelastic Collision: Momentum conserved, Kinetic Energy NOT conserved (max loss), objects stick ($e=0$)\n - 1D: $m_1u_1 + m_2u_2 = (m_1 + m_2)V$\n- Coefficient of Restitution (e): $e = -\frac{(v_2 - v_1)}{(u_2 - u_1)}$ or $e = \frac{\text{relative speed of separation}}{\text{relative speed of approach}}$\n- Bouncing Ball: $e = \sqrt{h_{rebound}/h_{drop}}$

2-Minute Revision

Collisions are brief interactions where objects exchange momentum and energy. The cornerstone of collision analysis is the Conservation of Linear Momentum, which states that the total momentum of an isolated system remains constant before and after any collision ($m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2$).

This applies universally to all collision types. \n\nCollisions are categorized by their kinetic energy behavior: \n1. Elastic Collisions: Both momentum and total kinetic energy are conserved. The coefficient of restitution ($e$) is 1.

A key special case is when two equal masses collide elastically in 1D, they exchange velocities. \n2. Inelastic Collisions: Momentum is conserved, but kinetic energy is not; some is lost to heat, sound, or deformation.

Here, $0 < e < 1$. \n3. Perfectly Inelastic Collisions: Objects stick together after impact, moving as a single unit. Momentum is conserved, but kinetic energy loss is maximal. The coefficient of restitution ($e$) is 0.

\n\nImpulse ($J = \Delta p$) quantifies the effect of force over time. For 2D collisions, momentum conservation must be applied component-wise. Remember to convert units and maintain sign conventions for vector quantities.

5-Minute Revision

Collisions are interactions where objects exert strong forces over a short time, changing their motion. The fundamental principle is the Conservation of Linear Momentum: for an isolated system, total momentum before equals total momentum after ($m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2$).

Remember momentum is a vector, so directions (signs) are crucial, especially in 1D, and components in 2D.\n\nTypes of Collisions:\n1. Elastic: Momentum and *kinetic energy* are conserved. $e=1$.

Example: Ideal billiard balls. For 1D elastic collisions, specific formulas exist for final velocities: $v_1 = \frac{(m_1 - m_2)u_1 + 2m_2u_2}{m_1 + m_2}$ and $v_2 = \frac{(m_2 - m_1)u_2 + 2m_1u_1}{m_1 + m_2}$.

A common special case is equal masses exchanging velocities.\n2. Inelastic: Momentum conserved, but kinetic energy is *not* conserved (some is lost). $0 < e < 1$. Example: Car crash.\n3. Perfectly Inelastic: Objects stick together after impact.

Momentum conserved, maximum kinetic energy loss. $e=0$. Example: Bullet embedding in a block. The common final velocity $V = \frac{m_1u_1 + m_2u_2}{m_1 + m_2}$.\n\nCoefficient of Restitution (e): Defined as $e = -\frac{(v_2 - v_1)}{(u_2 - u_1)}$.

It's the ratio of relative speed of separation to relative speed of approach. For a ball bouncing off a stationary surface, $e = \sqrt{h_{rebound}/h_{drop}}$.\n\nImpulse: The change in momentum, $\vec{J} = \Delta \vec{p}$.

Also, $\vec{J} = \vec{F}_{avg} \Delta t$. This links force and time to momentum change.\n\nWorked Example: A $0.2\,\text{kg}$ ball moving at $20\,\text{m/s}$ hits a wall and rebounds at $15\,\text{m/s}$.

What is the impulse imparted by the wall?\n* Initial momentum $p_i = (0.2\,\text{kg})(20\,\text{m/s}) = 4\,\text{kg\cdot m/s}$.\n* Final momentum $p_f = (0.2\,\text{kg})(-15\,\text{m/s}) = -3\,\text{kg\cdot m/s}$ (taking initial direction as positive).

\n* Impulse $J = p_f - p_i = -3 - 4 = -7\,\text{kg\cdot m/s}$. The magnitude is $7\,\text{N\cdot s}$, and the direction is opposite to the initial motion of the ball.

Prelims Revision Notes

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Definition: — Collision is a brief, intense interaction between bodies leading to momentum and energy exchange.\n2. Linear Momentum: $\vec{p} = m\vec{v}$. It's a vector. Units: kg\cdot m/s.\n3. Impulse: $\vec{J} = \Delta \vec{p} = \vec{F}_{avg} \Delta t$. Units: N\cdot s or kg\cdot m/s. Impulse is the change in momentum.\n4. Conservation of Linear Momentum: For an isolated system (no external forces), total momentum is always conserved in ALL types of collisions. $m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2$. For 2D, apply component-wise ($x$ and $y$ components separately).\n5. Types of Collisions (based on Kinetic Energy):\n * Elastic Collision: Momentum conserved, Kinetic Energy conserved. $e=1$. Objects regain shape. Relative speed of separation = relative speed of approach. \n * 1D Elastic Collision Formulas: \n $v_1 = \frac{(m_1 - m_2)u_1 + 2m_2u_2}{m_1 + m_2}$ \n $v_2 = \frac{(m_2 - m_1)u_2 + 2m_1u_1}{m_1 + m_2}$ \n * Special Case: If $m_1=m_2$, then $v_1=u_2$ and $v_2=u_1$ (velocities exchange).\n * Special Case: If $m_2 \gg m_1$ and $u_2=0$, then $v_1 \approx -u_1$ and $v_2 \approx 0$ (lighter object rebounds, heavier remains at rest).\n * Inelastic Collision: Momentum conserved, Kinetic Energy NOT conserved (some lost). $0 < e < 1$. Objects may deform.\n * Perfectly Inelastic Collision: Momentum conserved, Kinetic Energy NOT conserved (maximum loss). $e=0$. Objects stick together and move as one. \n * Common final velocity $V = \frac{m_1u_1 + m_2u_2}{m_1 + m_2}$.\n6. Coefficient of Restitution (e): $e = -\frac{(v_2 - v_1)}{(u_2 - u_1)} = \frac{\text{relative speed of separation}}{\text{relative speed of approach}}$. For a ball bouncing off the ground, $e = \sqrt{h_{rebound}/h_{drop}}$.\n7. Key Problem-Solving Steps:\n * Identify collision type.\n * Draw diagram, define positive directions/axes.\n * Apply conservation of momentum (always).\n * Apply kinetic energy conservation (if elastic) or 'e' equation (if inelastic/perfectly inelastic).\n * Solve simultaneous equations. Watch units and signs.

Vyyuha Quick Recall

MEPI: Momentum Everywhere, Perfectly Inelastic sticks, Elastic Energy too. (Momentum is conserved in Every collision. Perfectly Inelastic collisions mean objects stick together. Elastic collisions conserve Energy, in addition to momentum.)