CH8-EXAM-2-Momentum, Impulse, and Conservation: Study Notes for Physics

CHAPTER 8: Momentum, Impulse, and Conservation

Introduction to New Physical Quantities

In this chapter, we introduce new physical quantities essential for analyzing motion and interactions in physics: momentum and impulse. These concepts are foundational for understanding collisions and multi-particle systems, and are closely related to Newton's laws of motion.

• Translational Momentum (often called linear momentum) is a measure of an object's motion, defined as the product of its mass and velocity.

• Impulse quantifies the effect of a force acting over a time interval, leading to a change in momentum.

• The impulse-momentum theorem connects impulse and momentum, analogous to the work-energy theorem.

• The principle of conservation of momentum is introduced, with emphasis on the conditions under which it applies.

• Analysis is extended to multi-particle systems and collisions.

Momentum

Definition and Properties

Momentum is a vector quantity that describes the motion of a particle. It is denoted by \( \vec{p} \) and defined as:

• Formula: $\vec{p} = m \vec{v}$ where m is mass and \( \vec{v} \) is velocity.

• Units: kg·m/s (no special name).

• Vector Equation: Momentum has components in each coordinate direction: $p_x = m v_x$ $p_y = m v_y$

• Kinetic Energy in Terms of Momentum: $K = \frac{p^2}{2m}$ This relates kinetic energy to momentum.

Example: A 2 kg object moving at 3 m/s has momentum $p = 2 \times 3 = 6$ kg·m/s.

Newton's Second Law in Terms of Momentum

Formulation and Application

Newton's second law can be expressed using momentum, providing a more general framework for analyzing forces and motion:

• General Form: $\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$

• This is closer to Newton's original formulation in Principia.

• Component Form: $F_{\text{net},x} = \frac{dp_x}{dt}$ $F_{\text{net},y} = \frac{dp_y}{dt}$

Example: If a 5 kg object’s momentum changes from 10 to 20 kg·m/s in 2 seconds, the net force is $F_{\text{net}} = \frac{20-10}{2} = 5$ N.

Impulse

Definition and Calculation

Impulse is the effect of a force acting over a time interval, resulting in a change in momentum. It is a vector quantity with the same direction as the force.

• Formula for Constant Force: $\vec{J} = \vec{F} (t_f - t_i) = \vec{F} \Delta t$

• Units: N·s = kg·m/s (same as momentum).

• Impulse is the area under a force vs. time graph.

Example: A 10 N force acts for 3 s: $J = 10 \times 3 = 30$ N·s.

Impulse-Momentum Theorem

Relationship Between Impulse and Momentum

The impulse-momentum theorem states that the total impulse delivered to an object equals its change in momentum.

• General Form: $\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}$

• Impulse: $\vec{J} = \vec{F}_{\text{net}} \Delta t$

• Total Change in Momentum: $\vec{J}_{\text{tot}} = \Delta \vec{p} = \vec{p}_2 - \vec{p}_1 = \vec{F}_{\text{net}} \Delta t$

• Component Form: $J_{\text{tot},x} = p_{2,x} - p_{1,x} = F_{\text{net},x} \Delta t$ $J_{\text{tot},y} = p_{2,y} - p_{1,y} = F_{\text{net},y} \Delta t$

Example: If a 2 kg object’s velocity changes from 1 m/s to 4 m/s in 2 s, $\Delta p = 2 \times (4-1) = 6$ kg·m/s, so $J = 6$ kg·m/s.

Work-Energy and Impulse-Momentum Theorems

Comparison and Application

Both the work-energy theorem and impulse-momentum theorem are integral formulations of Newton's second law, relating force to changes in energy and momentum, respectively.

• Work-Energy Theorem: $\int \vec{F}_{\text{net}} \cdot d\vec{s} = \Delta K$ (Scalar equation: $W_{\text{tot}} = \Delta K$)

• Impulse-Momentum Theorem: $\int \vec{F}_{\text{net}} dt = \Delta \vec{p}$ (Vector equation: $\vec{J}_{\text{tot}} = \Delta \vec{p}$)

Additional info: These theorems provide alternative methods for solving problems involving forces, energy, and momentum.