Collisions — Revision Notes

⚡ 30-Second Revision

Linear Momentum: — $\vec{p} = m\vec{v}$ (vector quantity)
Impulse: — $\vec{J} = \Delta\vec{p} = \vec{F}_{avg}\Delta t$
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)
Kinetic Energy: — $KE = \frac{1}{2}mv^2$
Elastic Collision: — Momentum conserved, KE conserved, $e=1$.

- 1D: $u_1 - u_2 = -(v_1 - v_2)$ (Relative speed of approach = Relative speed of separation)

Inelastic Collision: — Momentum conserved, KE *not* conserved, $0 \le e < 1$.
Perfectly Inelastic Collision: — Objects stick together, $e=0$. Max KE loss.

- 1D: $m_1u_1 + m_2u_2 = (m_1 + m_2)V$

Coefficient of Restitution (e): — $e = \frac{|v_2 - v_1|}{|u_1 - u_2|}$ (Ratio of relative speed of separation to approach)
Rebound Height: — $h = e^2H$ (for a ball dropped from H)

2-Minute Revision

Collisions are brief, intense interactions where objects exchange momentum. The bedrock principle is the conservation of linear momentum: for an isolated system, total momentum before equals total momentum after, always.

This is a vector sum. Impulse is the change in momentum, caused by the force over the collision time. Collisions are classified by kinetic energy conservation. In an elastic collision, both momentum and kinetic energy are conserved.

For 1D elastic collisions, the relative speed of approach equals the relative speed of separation. In an inelastic collision, momentum is conserved, but kinetic energy is lost (converted to heat, sound, deformation).

A perfectly inelastic collision is when objects stick together, resulting in maximum kinetic energy loss. The coefficient of restitution (e) quantifies elasticity: $e=1$ for elastic, $e=0$ for perfectly inelastic, and $0 < e < 1$ for other inelastic collisions.

For a bouncing ball, rebound height $h = e^2H$. Remember to use vector notation and signs consistently for 1D problems.

5-Minute Revision

Collisions are fundamental interactions where objects exert strong forces on each other over short durations, leading to changes in their motion. The Impulse-Momentum Theorem ($\vec{J} = \Delta\vec{p}$) is key, stating that the impulse (force $\times$ time) equals the change in momentum.

The most crucial principle is the Conservation of Linear Momentum: for any isolated system, the total linear momentum ($\sum m\vec{v}$) before a collision is equal to the total linear momentum after.

This holds true for all types of collisions.

Collisions are primarily categorized by Kinetic Energy (KE) conservation:

1
Elastic Collisions: — Both linear momentum and kinetic energy are conserved. These are ideal collisions, often approximated by hard, bouncy objects. For 1D elastic collisions, a key result is that the relative speed of approach equals the relative speed of separation ($u_1 - u_2 = -(v_1 - v_2)$). Special cases include equal masses exchanging velocities, or a light mass rebounding from a heavy stationary mass with nearly its initial speed.
2
Inelastic Collisions: — Linear momentum is conserved, but kinetic energy is *not*. Some KE is converted into heat, sound, or deformation energy. Most real-world collisions are inelastic.
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Perfectly Inelastic Collisions: — A special case where objects stick together after impact and move as a single combined mass. This results in the maximum possible loss of kinetic energy. The common final velocity $V = \frac{m_1u_1 + m_2u_2}{m_1 + m_2}$.

The Coefficient of Restitution (e) is a dimensionless measure of elasticity: $e = \frac{|v_2 - v_1|}{|u_1 - u_2|}$.

$e=1$: Perfectly elastic collision.
$e=0$: Perfectly inelastic collision.
$0 < e < 1$: Inelastic collision.

For a ball dropped from height $H$ rebounding to $h$, $e = \sqrt{h/H}$, so $h = e^2H$.

Example: A $1,\text{kg}$ ball moving at $5,\text{m/s}$ collides elastically with a stationary $1,\text{kg}$ ball. Since masses are equal and it's elastic, they exchange velocities. The first ball stops ($v_1=0$), and the second ball moves at $5,\text{m/s}$ ($v_2=5,\text{m/s}$).

Momentum: $1(5)+1(0) = 5$. After: $1(0)+1(5) = 5$. KE: $\frac{1}{2}(1)(5^2) = 12.5$. After: $\frac{1}{2}(1)(0^2) + \frac{1}{2}(1)(5^2) = 12.5$. Both conserved. If it were perfectly inelastic, $V = \frac{1(5)+1(0)}{1+1} = 2.

5, ext{m/s}$. KE initial$12.5, ext{J}$, KE final$\frac{1}{2}(2)(2.5^2) = 6.25, ext{J}$. KE is lost.

Prelims Revision Notes

1
Linear Momentum ($\vec{p}$): — Product of mass and velocity ($m\vec{v}$). Vector quantity. Unit: kg m/s or Ns.
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Impulse ($\vec{J}$): — Change in momentum ($\Delta\vec{p}$). Also, $\vec{J} = \int \vec{F} \, dt = \vec{F}_{avg}\Delta t$. Impulse is a vector.
3
Conservation of Linear Momentum: — For an isolated system (no net external force), total momentum before collision = total momentum after collision. $m_1\vec{u}_1 + m_2\vec{u}_2 = m_1\vec{v}_1 + m_2\vec{v}_2$. This is ALWAYS true for collisions in an isolated system.
4
Kinetic Energy ($KE$): — Energy of motion ($\frac{1}{2}mv^2$). Scalar quantity.
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Types of Collisions:

* Elastic Collision: * Linear momentum conserved. * Kinetic energy conserved. * Total energy conserved. * Coefficient of restitution $e=1$. * Relative speed of approach = Relative speed of separation: $u_1 - u_2 = -(v_1 - v_2)$.

* Special cases (1D, $u_2=0$): * $m_1 = m_2$: $v_1=0, v_2=u_1$ (velocities exchanged). * $m_1 \ll m_2$: $v_1 \approx -u_1, v_2 \approx 0$ (light rebounds, heavy stays). * $m_1 \gg m_2$: $v_1 \approx u_1, v_2 \approx 2u_1$ (heavy continues, light moves with double speed).

* Inelastic Collision: * Linear momentum conserved. * Kinetic energy *not* conserved ($KE_{initial} > KE_{final}$). Energy lost as heat, sound, deformation. * Total energy (including non-mechanical forms) is conserved.

* Coefficient of restitution $0 \le e < 1$. * Perfectly Inelastic Collision: * Objects stick together after collision, moving with a common final velocity $V$. * Linear momentum conserved: $m_1u_1 + m_2u_2 = (m_1 + m_2)V$.

* Maximum possible loss of kinetic energy. * Coefficient of restitution $e=0$.

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Coefficient of Restitution (e): — $e = \frac{\text{Relative speed of separation}}{\text{Relative speed of approach}} = \frac{|v_2 - v_1|}{|u_1 - u_2|}$.
2
Rebound Height: — For a ball dropped from height $H$ and rebounding to height $h$, $e = \sqrt{h/H}$, so $h = e^2H$.
3
Ballistic Pendulum: — A common problem combining perfectly inelastic collision (momentum conservation) and subsequent swing (mechanical energy conservation).

Vyyuha Quick Recall

MICE KEPT: Momentum Is Conserved for Every collision. Kinetic Energy Preserves Totally (only for Elastic).