Physics I: Study Guide – Energy, Power, Impulse, and Momentum

Energy and Power

Introduction to Energy and Power

Energy and power are fundamental concepts in physics, describing the ability to do work and the rate at which work is performed. Understanding these concepts is essential for analyzing physical systems and solving problems involving motion, forces, and transformations.

• Energy is the capacity to do work. It exists in various forms, such as kinetic, potential, thermal, and more.

• Power is the rate at which energy is transferred or converted. It is measured in watts (W).

Key Equations

• Kinetic Energy: $K = \frac{1}{2}mv^2$

• Potential Energy (gravitational): $U = mgh$

• Work: $W = F \cdot d \cdot \cos\theta$

• Power: $P = \frac{W}{t}$

Examples and Applications

• Calculating the energy change when an object moves between two heights.

• Determining the power output of a machine lifting a weight over time.

Additional info: Energy conservation is a key principle—energy can be transformed but not created or destroyed in an isolated system.

Impulse and Momentum

Impulse and Momentum Concepts

Momentum and impulse are central to understanding how forces affect the motion of objects, especially during collisions and interactions.

• Momentum (p): The product of an object's mass and velocity, $p = mv$.

• Impulse (J): The change in momentum resulting from a force applied over a time interval, $J = F \Delta t$.

• Impulse-Momentum Theorem: $J = \Delta p$

Key Equations

• Impulse: $J = \int F\,dt$

• Conservation of Momentum: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$ (for isolated systems)

Examples and Applications

• Calculating the change in velocity of an object after a force acts for a certain time.

• Analyzing collisions (elastic and inelastic) using conservation of momentum.

Additional info: Impulse is equal to the area under a force vs. time graph.

Work and Energy Relationships

Work-Energy Principle

The work done by all forces acting on an object equals the change in its kinetic energy. This principle is useful for solving problems involving forces and motion.

• Work-Energy Theorem: $W_{\text{net}} = \Delta K = K_f - K_i$

• Work can be positive (energy added) or negative (energy removed).

Example

• Finding the final speed of an object after work is done by a net force.

Potential Energy and Equilibrium

Potential Energy Curves and Stability

Potential energy diagrams help visualize the stability of equilibrium points and the motion of particles in conservative force fields.

• Stable Equilibrium: Small displacements result in forces that return the object to equilibrium.

• Unstable Equilibrium: Small displacements result in forces that move the object further from equilibrium.

Key Points

• At stable equilibrium, potential energy is at a minimum.

• At unstable equilibrium, potential energy is at a maximum.

Collisions and Conservation Laws

Types of Collisions

Collisions are analyzed using conservation of momentum and, in some cases, conservation of kinetic energy.

• Elastic Collision: Both momentum and kinetic energy are conserved.

• Inelastic Collision: Momentum is conserved, but kinetic energy is not (objects may stick together).

Example Problem

• Two cars collide and stick together. Use conservation of momentum to find their final velocity.

Spring Potential Energy

Hooke's Law and Elastic Potential Energy

Springs store energy when compressed or stretched, described by Hooke's Law and elastic potential energy.

• Hooke's Law: $F = -kx$

• Elastic Potential Energy: $U = \frac{1}{2}kx^2$

Example

• Calculating the energy stored in a spring stretched by a certain amount.

Summary Table: Key Quantities and Equations

Quantity	Symbol	Equation	SI Unit
Kinetic Energy	K	$K = \frac{1}{2}mv^2$	Joule (J)
Potential Energy (gravity)	U	$U = mgh$	Joule (J)
Elastic Potential Energy	U	$U = \frac{1}{2}kx^2$	Joule (J)
Momentum	p	$p = mv$	kg·m/s
Impulse	J	$J = F\Delta t$	kg·m/s
Power	P	$P = \frac{W}{t}$	Watt (W)

Conceptual Questions and Applications

• How does friction affect energy conservation in a system?

• What is the relationship between force, work, and energy?

• How do you determine the energy lost in an inelastic collision?

• Where are various forms of energy stored in a system?

Additional info: For all calculations, use SI units unless otherwise specified. Always check if energy is conserved or if external forces (like friction) are present.