Linear Momentum and Collisions: Conservation Principles, Impulse, and Center of Mass

Linear Momentum

Definition and Properties

Linear momentum is a fundamental concept in physics, describing the quantity of motion an object possesses. It is defined for a mass m moving with velocity v as:

• Momentum (\( \vec{p} \)): \( \vec{p} \equiv m \vec{v} \)

• Vector Nature: The direction of \( \vec{p} \) is the same as the direction of \( \vec{v} \).

• Units: \( [\vec{p}] = \mathrm{kg \cdot m/s} \) (no special name).

• Total Momentum (for multiple masses): \( \vec{p}_{\mathrm{tot}} = \sum_i m_i \vec{v}_i \)

Note: The symbol "p" is used for momentum because "m" is reserved for mass.

Conservation of Momentum

Momentum is a conserved quantity in physics, meaning it cannot be created or destroyed, only transferred between objects. For a system isolated from external forces, the total momentum remains constant:

• Conservation Law: \( \vec{p}_{\mathrm{tot, before}} = \vec{p}_{\mathrm{tot, after}} \)

• Especially useful for analyzing collisions between particles.

• Similar in principle to conservation of energy.

Types of Collisions

Elastic and Inelastic Collisions

• Elastic Collision: Both momentum and kinetic energy (KE) are conserved. Example: Collisions between air molecules.

• Inelastic Collision: Momentum is conserved, but some KE is lost to thermal energy, sound, etc. All macroscopic collisions are inelastic to some degree.

• Perfectly Inelastic Collision: Colliding objects stick together after the collision.

1D Collisions

• In one dimension, vector direction is indicated by sign: (+) right, (–) left.

• Notation: In 1D, "v" can be positive or negative (velocity), not always positive (speed).

Example: Perfectly Inelastic Collision

Object A (mass \( m_A \)) with velocity \( v \) collides with object B (mass \( m_B \)), initially at rest. After collision, they stick together with velocity \( v' \):

• Conservation of momentum: $m_A v = (m_A + m_B) v'$ $\Rightarrow v' = \frac{m_A}{m_A + m_B} v$

• Since \( \frac{m_A}{m_A + m_B} < 1 \), \( v' < v \).

Example: Recoil of a Gun

A gun (mass \( M \)) fires a bullet (mass \( m \)) with velocity \( v_b \). The gun recoils with velocity \( v_G \):

• Conservation of momentum: $0 = M v_G + m v_b$ $\Rightarrow v_G = -\frac{m}{M} v_b$

• Numerical example: \( v_b = 500 \) m/s, \( m = 0.01 \) kg, \( M = 3 \) kg $v_G = -\frac{0.01}{3} \times 500 \approx -1.7$ m/s

• This principle explains rocket propulsion: expelling mass backward propels the rocket forward.

Impulse and Momentum Change

Impulse

• Impulse (\( J \)): The product of net force and the time interval over which it acts. $J = F_{\mathrm{net}} \Delta t$ (for constant force)

• For variable force: $J = \int_{t_i}^{t_f} F_{\mathrm{net}} dt$

• Impulse equals the change in momentum: $J = \Delta p = m (v_f - v_i)$

Example: Bat Hits Baseball

• \( m = 0.30 \) kg, \( v_i = -42 \) m/s, \( v_f = +80 \) m/s, \( \Delta t = 0.010 \) s

• Impulse: $J = 0.30 \times (80 - (-42)) = 0.30 \times 122 = 36.6 \approx 37$ kg·m/s

• Average force: $F_{\mathrm{avg}} = \frac{J}{\Delta t} = \frac{37}{0.010} = 3700$ N

Proof of Conservation of Momentum

Newton's Third Law and Collisions

• When two objects collide, each exerts an equal and opposite force on the other for the same duration (Newton's Third Law).

• Each object receives an impulse of equal magnitude but opposite direction.

• Thus, the total change in momentum is zero: $\Delta p_A + \Delta p_B = 0$ $\Rightarrow p_A + p_B = \text{constant}$

• Momentum is conserved if all forces are internal (system is isolated).

Ballistic Pendulum: Conservation of Energy and Momentum

Application Example

The ballistic pendulum is used to measure the speed of a bullet by combining conservation of momentum and energy principles.

• Bullet (mass \( m \), velocity \( v_1 \)) embeds in block (mass \( M \)), initially at rest.

• Immediately after collision: $m v_1 = (M + m) v_2$

• Block+bullet swing upward to height \( h \): $\frac{1}{2} (M + m) v_2^2 = (M + m) g h$

• Solving these equations yields the initial bullet velocity \( v_1 \).

Elastic Collisions in One Dimension

Conservation Laws

• For elastic collisions, both momentum and kinetic energy are conserved:

$m_A v_A + m_B v_B = m_A v_A' + m_B v_B'$ $\frac{1}{2} m_A v_A^2 + \frac{1}{2} m_B v_B^2 = \frac{1}{2} m_A v_A'^2 + \frac{1}{2} m_B v_B'^2$

• Relative velocity reverses in elastic collisions: $v_B' - v_A' = -(v_B - v_A)$

Example: Elastic Collision with Mass Ratio

• \( m_A = 10m \), \( v_A \), \( m_B = m \), \( v_B = 0 \)

• Momentum: $10 v_A = 10 v_A' + v_B'$

• Relative velocity: $v_B' - v_A' = v_A$

• Solving yields: $v_A' = \frac{9}{11} v_A$ $v_B' = \frac{20}{11} v_A$

Center of Mass

Definition and Calculation

• The center of mass (c.m.) of a system of particles is the weighted average of their positions:

$\vec{R} = \frac{\sum_i m_i \vec{r}_i}{M}$ where $M = \sum_i m_i$

• For coordinates: $X = \frac{\sum_i m_i x_i}{M}$, $Y = \frac{\sum_i m_i y_i}{M}$, $Z = \frac{\sum_i m_i z_i}{M}$

Example: Four-Mass System

• Four masses at corners of a square (edge length \( d \)): three masses \( m \), one mass \( 3m \).

• Coordinates: $X = 0.67d$, $Y = 0.33d$ (closer to heavier mass).

Motion of the Center of Mass

• The center of mass moves as if all external forces act on a single particle of mass \( M \) at \( \vec{R} \):

$\vec{F}_{\mathrm{net}} = M \vec{A}$ where $\vec{A}$ is the acceleration of the center of mass.

• Internal forces cancel due to Newton's Third Law; only external forces affect the center of mass motion.

• For an isolated system, the total momentum is conserved: $\vec{P}_{\mathrm{tot}} = M \vec{V}$ (\( \vec{V} \) is the velocity of the center of mass).

Appendix: Proof of Relative Velocity Reversal in Elastic Collisions

• From conservation of momentum and kinetic energy, it can be shown that:

$v_B' - v_A' = -(v_B - v_A)$

• This means the relative velocity of approach before collision is equal in magnitude and opposite in direction to the relative velocity of separation after collision.

Summary Table: Types of Collisions

Type of Collision	Momentum Conserved?	Kinetic Energy Conserved?	Example
Elastic	Yes	Yes	Air molecules colliding
Inelastic	Yes	No (some lost as heat, sound, etc.)	Car crash
Perfectly Inelastic	Yes	No (objects stick together)	Bullet embedding in block

Additional info: The notes above include expanded explanations, examples, and equations for clarity and completeness, suitable for college-level physics students preparing for exams.