A complete guide to introductory university physics (mechanics). Covers kinematics, Newton's laws, work-energy, momentum, rotational motion, and oscillations. Built for Physics 1 / PHYA10 / PHYS 170.
Physics problems aren't solved by memorizing equations - they're solved by understanding what's happening physically, then translating that into math. Use this framework for every problem:
The IDEA Method
Identify - What type of problem is this? (kinematics, forces, energy, momentum?) What are you solving for?
Draw - Sketch the situation. Draw free body diagrams. Label all known and unknown quantities. Choose a coordinate system.
Execute - Write the relevant equations. Plug in values. Solve algebraically before substituting numbers when possible.
Assess - Check units. Check signs. Does the answer make physical sense? Check limiting cases (what happens when mass → 0, angle → 90°, etc.).
The #1 Physics Mistake
Jumping straight to equations without drawing a diagram. A free body diagram isn't optional - it's the single most important step. Students who skip it get the wrong answer 3x more often. Draw it every time, even when you think the problem is simple.
2. Kinematics (Motion in 1D and 2D)
Kinematics describes how things move without asking why. No forces, no masses - just position, velocity, acceleration, and time.
The Four Kinematic Equations
These apply ONLY when acceleration is constant:
Kinematic Equations (constant a)
v = v₀ + at
x = x₀ + v₀t + ½at²
v² = v₀² + 2a(x − x₀)
x = x₀ + ½(v₀ + v)t
Choosing the Right Equation
Missing Variable
Use This Equation
x (displacement)
v = v₀ + at
v (final velocity)
x = x₀ + v₀t + ½at²
t (time)
v² = v₀² + 2a(x − x₀)
a (acceleration)
x = x₀ + ½(v₀ + v)t
Projectile Motion (2D)
Separate horizontal and vertical components. They're independent - horizontal motion doesn't affect vertical, and vice versa.
Horizontal: a_x = 0, so x = x₀ + v₀ₓt (constant velocity)
Vertical: a_y = −g = −9.8 m/s², use kinematic equations
Launch angle θ: v₀ₓ = v₀ cos θ, v₀ᵧ = v₀ sin θ
At the peak: vᵧ = 0 (momentarily), vₓ unchanged
Quick Reference: Projectile on Level Ground
Time of flight: t = 2v₀ sin θ / g
Maximum height: H = v₀² sin²θ / 2g
Range: R = v₀² sin 2θ / g
Maximum range: θ = 45° (only on level ground with no air resistance)
Common Trap
Students often forget that at the top of a projectile's arc, the velocity is NOT zero - only the vertical component is zero. The object still has horizontal velocity. Similarly, acceleration is always g downward throughout the entire flight, including at the peak.
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Newton's laws explain why things move. This is where physics gets powerful - and where most students struggle.
The Three Laws
First Law (Inertia): An object stays at rest or in uniform motion unless a net external force acts on it. If ΣF = 0, then a = 0.
Second Law: ΣF = ma. The net force equals mass times acceleration. This is the workhorse equation of mechanics.
Third Law: For every action, there is an equal and opposite reaction. Forces come in pairs - but they act on DIFFERENT objects.
Drawing Free Body Diagrams
Isolate the object (draw it as a dot or simple shape)
Draw ALL forces acting on that object (not forces it exerts on others)
Label each force: Fg (gravity), FN (normal), Ff (friction), T (tension), Fa (applied)
Choose coordinate axes aligned with the direction of acceleration
Decompose forces into x and y components
Apply ΣFx = max and ΣFy = may separately
Common Forces
Gravity: Fg = mg (always down) Normal force: FN ⊥ to surface (NOT always equal to mg!) Tension: T along rope/string (massless rope → same T throughout) Spring: F = −kx (Hooke's law, restoring force) Kinetic friction: fk = μk · FN (opposes motion) Static friction: fs ≤ μs · FN (prevents motion, up to a maximum)
Third Law Trap
Third-law pairs act on DIFFERENT objects. If you push a wall, the wall pushes back on you - but these forces don't cancel because they're on different objects. They would only cancel if both acted on the same object. Never include third-law reaction forces in the same free body diagram.
4. Friction, Inclined Planes, and Circular Motion
Friction
Friction is not a single force - it has two types with different rules:
Static friction (fs): Prevents motion. Adjusts from 0 up to fs,max = μs · FN. It equals whatever force is needed to prevent sliding, up to the maximum.
Kinetic friction (fk): Acts during sliding. Constant magnitude fk = μk · FN. Always opposes the direction of motion.
Always: μs > μk (harder to start moving than to keep moving)
Inclined Planes
The key insight: tilt your coordinate axes to align with the slope.
x-axis: Along the incline (positive = up the slope, or whichever direction of acceleration)
y-axis: Perpendicular to the incline
Gravity components: Fg,x = mg sin θ (along slope), Fg,y = mg cos θ (into slope)
Normal force: FN = mg cos θ (NOT mg - this is the most common incline mistake)
Friction: f = μ · mg cos θ
Incline Problem Checklist
1. Draw the incline with the object on it
2. Tilt coordinate axes to match the slope
3. Decompose gravity: mg sin θ along slope, mg cos θ perpendicular
4. Normal force = mg cos θ (from ΣFy = 0)
5. Friction = μ × mg cos θ (if applicable)
6. Apply ΣFx = ma along the slope
Circular Motion
An object moving in a circle at speed v has centripetal acceleration directed toward the center:
Circular Motion
Centripetal acceleration: ac = v²/r (always toward center) Centripetal force: Fc = mv²/r (NOT a new force - it's the net force toward center) Period: T = 2πr/v Angular velocity: ω = 2π/T = v/r
Critical Concept
"Centripetal force" is NOT a separate force. It's the net force directed toward the center. On a banked curve, it's a component of the normal force. On a string, it's tension. In orbit, it's gravity. Never add "centripetal force" to a free body diagram - identify which real force provides the centripetal acceleration.
5. Work, Energy, and Conservation
Energy methods are often simpler than Newton's laws because they don't require knowing forces at every point - only at the start and end.
Work
Work Definitions
Work by constant force: W = F · d · cos θ (where θ is angle between F and displacement) Work by variable force: W = ∫F · dx Work-energy theorem: W_net = ΔKE = ½mv² − ½mv₀² Key: Only the component of force parallel to displacement does work
Types of Energy
Kinetic energy: KE = ½mv²
Gravitational PE: Ug = mgy (choose y = 0 at a convenient point)
Elastic PE: Us = ½kx² (spring potential energy)
Conservation of Energy
When only conservative forces (gravity, springs) do work:
When non-conservative forces (friction, air resistance) are present:
Work-Energy with Non-Conservative Forces
KE₁ + PE₁ + W_nc = KE₂ + PE₂
W_friction = −fk · d (always negative - removes energy from the system)
When to Use Energy vs. Forces
Use energy when: you care about speed/height at two points, the path doesn't matter, or forces vary along the path (like a roller coaster). Use forces (F = ma) when: you need acceleration, time, or forces at a specific point.
Power
Power
Average power: P = W/t = ΔE/t Instantaneous power: P = F · v (force times velocity) Units: Watts (W) = J/s
6. Momentum and Collisions
Momentum is conserved when no external forces act on a system. This makes it perfect for collisions, explosions, and any interaction between objects.
Special cases: Equal masses → v₁' = 0, v₂' = v₁ (complete transfer). Heavy hits light → both move forward. Light hits heavy → light bounces back.
2D Collisions
Apply conservation of momentum separately in x and y:
Σpx,before = Σpx,after
Σpy,before = Σpy,after
This gives you two equations. If the collision is elastic, KE conservation gives a third.
Center of Mass
Center of Mass
x_cm = Σ(mi · xi) / Σmi
v_cm = Σ(mi · vi) / Σmi = p_total / M_total Key: The center of mass of a system moves as if all external forces act on it at that point
7. Rotational Motion and Torque
Rotational mechanics mirrors translational mechanics - every linear concept has a rotational analog.
Rotational-Translational Analogs
Linear
Rotational
Relationship
Displacement x
Angular displacement θ
x = rθ
Velocity v
Angular velocity ω
v = rω
Acceleration a
Angular acceleration α
a = rα
Mass m
Moment of inertia I
I = Σmr²
Force F
Torque τ
τ = rF sin θ
F = ma
τ = Iα
-
KE = ½mv²
KE = ½Iω²
-
p = mv
L = Iω
-
Torque
Torque
τ = r × F = rF sin θ r = distance from axis to where force is applied θ = angle between r and F Moment arm = r sin θ = perpendicular distance from axis to line of force Sign: counterclockwise = positive, clockwise = negative (by convention)
Common Moments of Inertia
Shape
Axis
I
Point mass
Distance r from axis
mr²
Solid cylinder/disk
Through center
½mr²
Hollow cylinder
Through center
mr²
Solid sphere
Through center
⅖mr²
Hollow sphere
Through center
⅔mr²
Thin rod
Through center
1/12 · mL²
Thin rod
Through end
⅓mL²
Rolling Without Slipping
When an object rolls without slipping, the contact point has zero velocity relative to the ground:
Rolling Constraint
v_cm = Rω (velocity condition)
a_cm = Rα (acceleration condition) Total KE = ½mv²_cm + ½Iω² (translational + rotational)
Rolling Race
A solid sphere, hollow sphere, solid cylinder, and hollow cylinder all roll down the same incline from rest. Who wins? The solid sphere (smallest I/mr² ratio = ⅖). More of its energy goes to translation vs rotation. The ranking: solid sphere > solid cylinder > hollow sphere > hollow cylinder. Mass and radius don't matter - only the shape.
8. Simple Harmonic Motion
Any system with a restoring force proportional to displacement undergoes simple harmonic motion (SHM).
Mass-spring: T = 2π√(m/k), ω = √(k/m) Simple pendulum: T = 2π√(L/g), ω = √(g/L) (small angles only)
Energy in SHM
Total energy: E = ½kA² (constant throughout motion)
At maximum displacement (x = ±A): All PE, KE = 0, v = 0
At equilibrium (x = 0): All KE, PE = 0, v = v_max = Aω
At any point: ½kx² + ½mv² = ½kA²
Key Insight
The period of a mass-spring system does NOT depend on amplitude - a wide swing takes the same time as a narrow swing. Similarly, a pendulum's period doesn't depend on mass. These are counterintuitive facts that professors love to test.
9. Gravitation
Universal Gravitation
Force: F = GMm/r² (attractive, along line between centers) Gravitational PE: U = −GMm/r (negative! Zero at infinity) Surface gravity: g = GM/R² (R = planet radius) Orbital velocity: v = √(GM/r) Orbital period: T² = (4π²/GM)r³ (Kepler's third law) Escape velocity: v_esc = √(2GM/R)
Orbits
Circular orbit: Gravity provides centripetal force → GMm/r² = mv²/r
Higher orbit = slower: Both v and ω decrease with r (counterintuitive)
Total energy: E = −GMm/2r (negative for bound orbits)
Kepler's laws: Elliptical orbits, equal areas in equal times, T² ∝ r³
g vs. G
G = 6.674 × 10⁻¹¹ N·m²/kg² (universal gravitational constant - same everywhere). g = 9.8 m/s² (acceleration due to gravity at Earth's surface - varies by location). The connection: g = GM_Earth/R²_Earth.
10. Common Mistakes
Normal force = mg: Only true on flat, horizontal surfaces with no other vertical forces. On an incline, FN = mg cos θ. With an applied upward force, FN < mg.
Centripetal force on FBD: Never add "centripetal force" to a free body diagram. It's the net inward force provided by real forces (gravity, tension, normal).
Velocity = 0 means acceleration = 0: Wrong. A ball at the top of its arc has v = 0 but a = g downward. An object can have zero velocity and nonzero acceleration.
Heavier objects fall faster: In vacuum, all objects fall at the same rate (a = g regardless of mass). In air, terminal velocity depends on drag, which depends on shape.
Forgetting to decompose vectors: Forces and velocities have components. Always resolve into x and y (or parallel and perpendicular to the surface).
Sign errors: Define positive direction FIRST, then be consistent. If up is positive, gravity is −mg. If down the incline is positive, friction (opposing motion) is negative.
Using the wrong kinematic equation: Each equation is missing one variable. Identify what you don't have and don't need, then choose accordingly.
Confusing speed and velocity: Speed is a scalar (always positive). Velocity is a vector (has direction, can be negative). KE uses speed, momentum uses velocity.
Energy is not conserved (when it is): If only gravity acts, energy IS conserved. Many students default to kinematics when energy methods are simpler.
Applying rotational equations without moment of inertia: You can't use τ = Iα if you haven't calculated I first. Look up or derive I for the specific shape.
11. Exam Strategy
Before the Exam
Make a formula sheet (even if you get one provided - the act of making it is studying)
Practice problems by type, not by chapter. Do 5 incline problems in a row, then 5 energy problems.
Focus on problems you got wrong - redo them without looking at the solution
Time yourself on practice problems to build speed
During the Exam
Read the whole exam first. Start with the problem you're most confident about.
Draw diagrams for EVERY problem. This is worth at least 1-2 points in partial credit even if you can't solve it.
Write equations before numbers. Solve symbolically first, plug in numbers last. This reduces arithmetic errors and earns more partial credit.
Check units at every step. If your acceleration has units of m/s (not m/s²), something is wrong.
Estimate before calculating. If a car's acceleration comes out to 500 m/s², you made an error. Develop physical intuition.
Partial credit matters. Write the relevant equation, draw the free body diagram, show your setup - even if you can't finish the algebra.
Quick Topic Identification
Problem mentions...
Use...
Time, position, velocity, acceleration
Kinematics
Force, mass, tension, normal force
Newton's laws (F = ma)
Speed at two heights, spring, friction loss
Energy conservation
Collision, explosion, "before and after"
Momentum conservation
Spinning, angular, torque, rolling
Rotational dynamics (τ = Iα)
Orbit, planet, satellite
Gravitation (GMm/r²)
Spring oscillation, pendulum, period
SHM (ω = √(k/m))
Frequently Asked Questions
What are the 4 kinematic equations?
The four kinematic equations for constant acceleration are: (1) v = v₀ + at, (2) x = x₀ + v₀t + ½at², (3) v² = v₀² + 2a(x − x₀), and (4) x = x₀ + ½(v₀ + v)t. Each equation connects displacement, velocity, acceleration, and time - and each one is missing exactly one variable. To solve a kinematics problem, identify which variable you don't know and don't need, then pick the equation that doesn't contain it.
How do I draw a free body diagram?
Isolate the object - draw it as a simple dot or box. Identify every force acting ON the object (not forces the object exerts on others). Draw each force as an arrow starting from the object, pointing in the direction of the force. Label each arrow (Fg for gravity, FN for normal, Ff for friction, T for tension, Fa for applied). Choose a coordinate system - align one axis with the direction of acceleration. For inclined planes, tilt your axes to match the slope.
When do I use energy conservation vs. Newton's laws?
Use Newton's laws (F = ma) when you need to find forces, accelerations, or when the problem involves time explicitly. Use energy conservation when you only care about initial and final states and the path doesn't matter - especially for problems involving heights, speeds, and springs. Energy methods are often simpler for problems without friction. If non-conservative forces like friction do work, use the work-energy theorem.
What's the difference between elastic and inelastic collisions?
In elastic collisions, both momentum AND kinetic energy are conserved. Objects bounce off each other. In inelastic collisions, momentum is conserved but kinetic energy is NOT - some energy converts to heat, sound, or deformation. In perfectly inelastic collisions, objects stick together (maximum KE loss). To identify the type: if objects stick together, it's perfectly inelastic. If you're told "elastic," use both conservation laws.
How do I know when to use torque vs. force?
Use force (F = ma) when the object moves in a straight line (translational motion). Use torque (τ = Iα) when the object rotates around a fixed axis. Many problems require BOTH - for example, a ball rolling down a ramp involves translational F = ma along the slope AND rotational τ = Iα around the center. The key question: is something spinning? If yes, you need torque.
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