Key Formulas & Equations
Core Concepts
Force
A push or pull on an object resulting from its interaction with another object. Measured in Newtons (N). Forces cause objects to accelerate, decelerate, change direction, or deform.
Energy
The capacity to do work. It exists in many forms: kinetic (motion), potential (stored), thermal, chemical, nuclear, and electrical. Energy is conserved — it cannot be created or destroyed, only transformed.
Momentum
A vector quantity defined as the product of an object's mass and velocity (p = mv). In a closed system, total momentum is conserved before and after collisions.
Wave-Particle Duality
Quantum objects exhibit both wave-like and particle-like behavior. Light behaves as a wave (interference, diffraction) and as particles (photons in the photoelectric effect). Matter also has wave properties (de Broglie wavelength).
Thermodynamics
The study of heat, temperature, and energy transfer. The Four Laws: (0) Thermal equilibrium is transitive; (1) Energy is conserved; (2) Entropy of an isolated system never decreases; (3) Absolute zero is unattainable.
Electromagnetic Induction
A changing magnetic field induces an electromotive force (EMF) in a conductor. Discovered by Faraday: ε = -dΦ_B/dt. This principle underlies generators, transformers, and wireless charging.
Worked Examples
Example 1: Projectile Motion
Problem: A ball is thrown horizontally at 10 m/s from a 45 m cliff. How far does it land from the base?
Step 1: Find time to fall: h = (1/2)gt² → 45 = (1/2)(9.8)t²
Step 2: t² = 90/9.8 = 9.18 → t = 3.03 s
Step 3: Horizontal distance: x = v₀t = 10 × 3.03 = 30.3 m
Answer: The ball lands approximately 30.3 meters from the base.
Example 2: Newton's Second Law
Problem: A 5 kg object accelerates at 3 m/s². What net force acts on it?
Step 1: Apply F = ma
Step 2: F = (5 kg)(3 m/s²) = 15 N
Answer: The net force is 15 Newtons.
Example 3: Circuit Analysis
Problem: A 12 V battery is connected to a 4 Ω resistor. Find the current.
Step 1: Apply Ohm's Law: V = IR
Step 2: Solve for I: I = V/R = 12/4 = 3 A
Answer: The current is 3 Amperes.
Example 4: Gravitational Potential Energy
Problem: How much PE does a 2 kg book gain when lifted 3 m?
Step 1: PE = mgh = (2)(9.8)(3)
Step 2: PE = 58.8 J
Answer: The book gains 58.8 Joules of potential energy.
Practice Problems
Problem 1
A car accelerates from 0 to 20 m/s in 5 s. What is its acceleration?
Answer: a = Δv/Δt = 20/5 = 4 m/s²
Problem 2
What is the kinetic energy of a 0.5 kg ball moving at 10 m/s?
Answer: KE = (1/2)(0.5)(10)² = 25 J
Problem 3
Two charges of 2 μC and 3 μC are 0.5 m apart. Find the electrostatic force. (k = 9×10⁹ N·m²/C²)
Answer: F = k(q₁q₂)/r² = (9×10⁹)(2×10⁻⁶)(3×10⁻⁶)/(0.5)² = 0.216 N
Problem 4
A wave has frequency 50 Hz and wavelength 2 m. What is its speed?
Answer: v = fλ = 50 × 2 = 100 m/s
Problem 5
How much work is done pushing a box 5 m with a force of 20 N at 0° angle?
Answer: W = Fd·cos(0°) = 20 × 5 × 1 = 100 J
Problem 6
A 60 kg person stands on a scale in an elevator accelerating upward at 2 m/s². What does it read?
Answer: N = m(g+a) = 60(9.8+2) = 60 × 11.8 = 708 N