Momentum, Impulse, and Collisions (Sections 9.5–9.7): Study Notes for Physics for Life Sciences Lecture 15

Momentum and Impulse

Definitions and Principles

Momentum and impulse are fundamental concepts in physics that describe the motion of objects and how they change when forces are applied.

• Momentum (p): The product of an object's mass and its velocity. It is a vector quantity, meaning it has both magnitude and direction.

• Impulse (J): The effect of a force acting over a period of time, resulting in a change in momentum.

• Impulse-Momentum Theorem: The impulse delivered to an object is equal to the change in its momentum.

Formulas:

• Momentum: $\vec{p} = m \vec{v}$

• Impulse: $\vec{J} = \int \vec{F} \, dt = \Delta \vec{p}$

• Impulse-Momentum Theorem: $\vec{J} = \vec{p}_f - \vec{p}_i$

Example: When a bat strikes a ball, the force applied over the contact time changes the ball's momentum, demonstrating the impulse-momentum relationship.

Conservation Laws

General Principles

Conservation laws are foundational in physics, stating that certain quantities remain constant in isolated systems.

• Conservation of Momentum: In an isolated system (no net external force), the total momentum before an interaction equals the total momentum after.

Formula:

• Total momentum: $\vec{P} = \vec{p}_1 + \vec{p}_2 + \ldots$

• Conservation: $\vec{P}_f = \vec{P}_i$

Example: Two ice skaters push off from each other and move in opposite directions; their combined momentum remains constant if no external forces act.

Inelastic Collisions (Section 9.5)

Types and Characteristics

Collisions are classified based on how kinetic energy and momentum are conserved.

• Inelastic Collision: A collision in which kinetic energy is not conserved, but momentum is.

• Perfectly Inelastic Collision: The colliding objects stick together and move with a common final velocity.

Formula for Perfectly Inelastic Collisions:

• $(m_1 + m_2)v_f = m_1 v_{1i} + m_2 v_{2i}$

Example: A bullet embedding itself in a block of wood is a perfectly inelastic collision.

Sample Problem: Railroad Cars

• Two railroad cars of masses $2.0 \times 10^4$ kg and $4.0 \times 10^4$ kg collide and stick together.

• By applying conservation of momentum, the final velocity and the initial speed of the heavier car can be determined.

Calculation Steps:

• Set up the conservation equation: $(m_1 + m_2)v_f = m_1 v_{1i} + m_2 v_{2i}$

• Solve for the unknown initial velocity if other values are given.

Momentum and Collisions in Two Dimensions (Section 9.6)

Principles and Problem-Solving

When collisions occur in two dimensions, momentum must be conserved in each direction independently.

• Component Conservation: $\sum p_{x,\text{initial}} = \sum p_{x,\text{final}}$ and $\sum p_{y,\text{initial}} = \sum p_{y,\text{final}}$

• Each object’s momentum is broken into x and y components.

Example: A falcon swoops down at an angle and catches a pigeon flying horizontally. The combined velocity after the collision is found by conserving momentum in both x and y directions.

Calculation Steps:

• Find initial momentum components for each object.

• Set up conservation equations for x and y directions.

• Solve for the final velocity magnitude and direction using the Pythagorean theorem and trigonometry.

Formulas:

• $m_1 v_{1x,i} + m_2 v_{2x,i} = (m_1 + m_2) v_{fx}$

• $m_1 v_{1y,i} + m_2 v_{2y,i} = (m_1 + m_2) v_{fy}$

• Speed: $v_f = \sqrt{v_{fx}^2 + v_{fy}^2}$

• Angle: $\tan \theta = \frac{v_{fy}}{v_{fx}}$

Angular Momentum (Section 9.7)

Definitions and Conservation

Angular momentum is the rotational analog of linear momentum, important for systems involving rotation.

• Angular Momentum (L): For a rotating object, $L = I \omega$, where $I$ is the moment of inertia and $\omega$ is the angular velocity.

• SI Units: kg·m2/s

• Conservation of Angular Momentum: If no net external torque acts on a system, its total angular momentum remains constant.

Formulas:

• Angular momentum: $L = I \omega$

• Conservation: $I_i \omega_i = I_f \omega_f$

Example: A figure skater spins faster by pulling in their arms, reducing their moment of inertia and increasing angular velocity to conserve angular momentum.

Changing Moment of Inertia

• Moment of inertia depends on mass distribution relative to the axis of rotation.

• When the distribution changes (e.g., a person moves outward on a merry-go-round), the moment of inertia changes, affecting angular velocity.

Sample Problem: Merry-Go-Round

• When a person moves from the center to the edge, the system's moment of inertia increases, so the angular velocity decreases to conserve angular momentum.

• Period increases as $T = \frac{2\pi}{\omega}$.

Comparison Table: Rotational and Linear Dynamics

Rotational Dynamics	Linear Dynamics
Torque ($\tau$)	Force ($F$)
Moment of inertia ($I$)	Mass ($m$)
Angular velocity ($\omega$)	Velocity ($v$)
Angular momentum ($L = I\omega$)	Linear momentum ($p = mv$)

Summary: Problem-Solving Strategies

• Choose an isolated system or one isolated during the interaction.

• Draw diagrams showing the system before and after the event.

• Write conservation equations for momentum (linear or angular) in vector components.

• Check if the result is reasonable (e.g., direction, magnitude).

Key Concepts and Applications

• Impulse: Area under a force vs. time curve; changes momentum.

• Collisions: Inelastic (objects stick together), elastic (objects bounce apart), and two-dimensional cases.

• Angular Momentum: Conserved in absence of external torque; changes with moment of inertia.

Applications: Vehicle collisions, animal motion (e.g., falcons and prey), rotating systems (e.g., merry-go-rounds, skaters).