Accelerated Physics Final Exam Study Guide: Momentum, Energy, Gravity, and Relativity

Momentum and Impulse

Definition and Principles

Momentum and impulse are fundamental concepts in mechanics, describing the motion of objects and how they change under applied forces.

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

• Formula: $p = mv$

• Impulse (J): The change in momentum resulting from a force applied over a time interval.

• Formula: $J = F\Delta t$

• Impulse-Momentum Theorem: Impulse equals the change in momentum: $J = \Delta p$

• Increasing Impulse: Increase the force or the time interval over which the force acts.

• Reducing Force: Increasing the contact time during a collision (e.g., airbags, rolling with a punch) reduces the force experienced.

Example: In a car crash, if the stopping time is increased (e.g., by crumple zones or airbags), the force on passengers is reduced.

Collisions and Conservation of Momentum

Types of Collisions and Conservation Laws

Collisions are interactions between objects where momentum is transferred. The total momentum of a system is conserved in all collisions, but kinetic energy may or may not be conserved.

• Law of Conservation of Momentum: The total momentum before a collision equals the total momentum after the collision (in the absence of external forces).

• General Equation: $m_1v_{1i} + m_2v_{2i} = m_1v_{1f} + m_2v_{2f}$

• Elastic Collision: Both momentum and kinetic energy are conserved. Objects bounce off each other.

• Inelastic Collision: Only momentum is conserved; kinetic energy is not. Objects may deform or generate heat.

• Perfectly Inelastic Collision: Objects stick together after the collision and move as one mass.

Example: Two ice skaters push off from each other (elastic), or two cars lock bumpers and move together after a crash (perfectly inelastic).

Work, Energy, and Power

Definitions and Relationships

Work, energy, and power are central to understanding how forces cause motion and how energy is transferred or transformed.

• Work (W): The product of force and displacement in the direction of the force.

• Formula: $W = Fd$

• Potential Energy (PE): Stored energy due to position, especially in a gravitational field.

• Formula: $PE = mgh$

• Kinetic Energy (KE): Energy of motion.

• Formula: $KE = \frac{1}{2}mv^2$

• Power (P): The rate at which work is done.

• Formula: $P = \frac{W}{t}$

• Speed Effect: Kinetic energy increases with the square of speed; tripling speed increases KE by a factor of 9.

Example: Lifting a 3 kg object 2 m: $W = (3\,\text{kg})(9.8\,\text{m/s}^2)(2\,\text{m}) = 58.8\,\text{J}$

Gravity and Universal Gravitation

Newton's Law and Gravitational Effects

Gravity is the attractive force between masses. Newton's Law of Universal Gravitation quantifies this force and explains planetary motion and weightlessness in orbit.

• Newton's Law of Universal Gravitation: $F = G\frac{m_1m_2}{r^2}$

• Distance Effect: Doubling the distance between two masses reduces the gravitational force to one-fourth ($F/4$).

• Mass Effect: Doubling both masses increases the force by four times.

• Weightlessness: Astronauts in orbit experience free fall, appearing weightless because they are in continuous free fall around Earth.

• Surface Gravity: Shrinking a planet's radius increases surface gravity (if mass remains constant).

Example: If Earth's radius shrinks to one-third, surface gravity increases by a factor of nine.

Special Relativity

Postulates and Consequences

Special relativity describes the physics of objects moving at speeds close to the speed of light. It leads to surprising effects such as time dilation and length contraction.

• Speed of Light: The speed of light ($c$) is constant in all inertial frames of reference.

• Time Dilation: Moving clocks run slower compared to stationary ones.

• Formula: $t' = \frac{t}{\sqrt{1 - v^2/c^2}}$

• Length Contraction: Objects moving at high speeds contract in the direction of motion.

• Formula: $L' = L\sqrt{1 - v^2/c^2}$

• Mass–Energy Equivalence: Mass and energy are related by $E = mc^2$

Example: A spaceship traveling at 0.8c will experience time at 60% the rate of a stationary observer.

General Relativity

Gravity as Curved Spacetime

General relativity extends Newton's theory by describing gravity as the curvature of spacetime caused by mass and energy.

• Gravity: Not a force, but the result of curved spacetime.

• Mass Bends Spacetime: Massive objects cause spacetime to curve, and other objects follow these curves (geodesics).

• Time Dilation in Gravity: Time runs slower in stronger gravitational fields (near massive objects).

• Black Holes: Regions where spacetime curvature becomes extreme, and not even light can escape.

Example: Clocks on Earth's surface run slightly slower than clocks in orbit due to Earth's gravity.

Formula Reference Sheet

Quantity	Formula
Momentum	$p = mv$
Impulse	$J = F\Delta t$
Impulse-Momentum	$J = \Delta p$
Work	$W = Fd$
Potential Energy	$PE = mgh$
Kinetic Energy	$KE = \frac{1}{2}mv^2$
Power	$P = \frac{W}{t}$
Universal Gravitation	$F = G\frac{m_1m_2}{r^2}$
Mass–Energy Equivalence	$E = mc^2$
Time Dilation	$t' = \frac{t}{\sqrt{1 - v^2/c^2}}$
Length Contraction	$L' = L\sqrt{1 - v^2/c^2}$

Additional info: This study guide covers key topics from chapters on momentum, energy, gravity, and relativity, providing essential formulas and conceptual understanding for exam preparation.