Physics Problem-Solving: Tips and Common Formulas

TL;DR

• Physics problems follow a predictable pattern: identify what you know, figure out what you need, pick the right equation, solve
• Free-body diagrams are your best friend for force problems — draw them every single time
• Units are not optional — they tell you whether your answer makes sense and can help you find the right equation
• The same handful of formulas show up again and again — master those and you'll handle 80% of intro physics

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Physics is the subject where students say "I understand the concepts, but I can't solve the problems."

Sound familiar? You sit in lecture, the professor derives an equation, it makes sense. Then you try the homework and it's like the problems are in a different language from what you learned in class.

Here's the thing: physics problem-solving is a skill. Like any skill, it's learnable. You just need a systematic approach and enough practice.

This guide covers the most important formulas and problem-solving strategies for introductory physics. It won't teach you every topic — your textbook does that. It'll teach you how to approach problems so you can apply what you've learned.

The Universal Problem-Solving Method

Every physics problem, regardless of topic, follows this approach:

Step 1: Read and Visualize

Read the problem twice. The first time, get the overall picture. The second time, identify the specific information given.

Draw a picture. Seriously. A simple sketch clarifies the situation and helps you see what's happening physically, not just mathematically.

Step 2: List What You Know

Write down every quantity given in the problem with its symbol and units:

Given:
v₀ = 20 m/s (initial velocity)
a = -9.8 m/s² (acceleration due to gravity)
t = 3 s (time)
Find: y = ? (height)

Step 3: Identify What You Need

What does the problem ask you to find? Write it down with its symbol.

Step 4: Choose the Right Equation

Look at what you know and what you need. Pick the equation that connects them. This is where knowing your formulas matters — not memorizing them blindly, but understanding which formula applies when.

Step 5: Solve

Plug in numbers and solve. Keep units throughout the calculation — they'll tell you if you've made a mistake.

Step 6: Check

Does the answer make sense? Is the magnitude reasonable? Are the units correct? If a car's speed comes out to 50,000 m/s, you made an error somewhere.

Topic 1: Kinematics (Motion)

Kinematics describes how things move: position, velocity, acceleration.

The Big Five Kinematics Equations

For constant acceleration:

Equation	Variables	Use When
v = v₀ + at	v, v₀, a, t	Missing displacement
x = x₀ + v₀t + ½at²	x, v₀, a, t	Missing final velocity
v² = v₀² + 2a(x - x₀)	v, v₀, a, x	Missing time
x = x₀ + ½(v₀ + v)t	x, v₀, v, t	Missing acceleration
x = x₀ + vt - ½at²	x, v, a, t	Missing initial velocity

How to Choose the Right One

1. Write down your knowns and unknown
2. Find the equation that has all your knowns plus the one unknown
3. Each equation is missing one variable — match the missing variable to the one you DON'T need

Common Kinematics Problem: Projectile Motion

A ball is thrown upward at 15 m/s. How high does it go?

Known: v₀ = 15 m/s, a = -9.8 m/s², v = 0 m/s (at max height)
Find: Δy

Use: v² = v₀² + 2aΔy
0 = (15)² + 2(-9.8)(Δy)
0 = 225 - 19.6Δy
Δy = 225/19.6 = 11.5 m

Key projectile motion facts:

• At the top of the arc, vertical velocity = 0 (but horizontal velocity is unchanged)
• Time up = time down (for symmetric projectiles)
• Horizontal and vertical motion are INDEPENDENT — analyze them separately
• Horizontal: no acceleration (constant velocity)
• Vertical: acceleration = g = 9.8 m/s² downward

Topic 2: Forces (Newton's Laws)

Newton's Three Laws

First Law (Inertia): An object stays at rest or in constant motion unless acted on by a net force.

Second Law: F_net = ma (the net force on an object equals its mass times its acceleration)

Third Law: For every action force, there's an equal and opposite reaction force.

Free-Body Diagrams (DRAW THESE)

A free-body diagram shows all forces acting on a single object as arrows. Every force problem should start with one.

Common forces to include:

• Weight (W or F_g): mg, always points DOWN
• Normal force (N or F_N): Perpendicular to the surface, pushes AWAY from surface
• Friction (f): Parallel to surface, opposes motion
  • Static friction: f_s ≤ μ_sN (object not moving)
  • Kinetic friction: f_k = μ_kN (object sliding)
• Tension (T): Along the string/rope, pulls toward the string
• Applied force (F_app): Whatever's pushing or pulling

Solving Force Problems

1. Draw a free-body diagram
2. Choose a coordinate system (usually x along the direction of motion, y perpendicular)
3. Break forces into x and y components
4. Apply Newton's Second Law in each direction:
   • ΣF_x = ma_x
   • ΣF_y = ma_y
5. Solve the system of equations

Example: Block on a Ramp

A 5 kg block slides down a 30° frictionless ramp. What's its acceleration?

Forces: Weight (mg down), Normal force (perpendicular to ramp)

Choose axes: x along the ramp (down positive), y perpendicular to ramp

x-direction: mg sin(30°) = ma
a = g sin(30°) = 9.8 × 0.5 = 4.9 m/s²

y-direction: N - mg cos(30°) = 0
N = mg cos(30°) = 5(9.8)(0.866) = 42.4 N

Ramp problems tip: Weight components on a ramp are always mg sinθ (along the ramp) and mg cosθ (perpendicular to the ramp).

Topic 3: Energy

Key Energy Equations

Kinetic Energy: KE = ½mv²

Gravitational Potential Energy: PE = mgh

Work: W = Fd cos(θ) (where θ is the angle between force and displacement)

Work-Energy Theorem: W_net = ΔKE

Conservation of Energy: KE₁ + PE₁ = KE₂ + PE₂ (no friction)

When to Use Energy vs. Forces

Use energy when:

• The problem asks about speed at a different position
• You don't need to know the acceleration
• The path is curved or complex (energy doesn't care about the path)

Use forces when:

• The problem asks about acceleration
• You need to find a specific force value
• You need to analyze what's happening at a specific moment

Example: Conservation of Energy

A roller coaster starts at 20 m high with zero speed. How fast is it going at 5 m high? (No friction)

KE₁ + PE₁ = KE₂ + PE₂
0 + mgh₁ = ½mv² + mgh₂

Mass cancels:
gh₁ = ½v² + gh₂
9.8(20) = ½v² + 9.8(5)
196 = ½v² + 49
½v² = 147
v² = 294
v = 17.1 m/s

Notice how the mass doesn't matter? That's a beautiful feature of energy conservation.

Topic 4: Momentum

Momentum: p = mv

Impulse: J = FΔt = Δp

Conservation of Momentum: In a system with no external forces, total momentum before = total momentum after

Collisions

Elastic collision (objects bounce off each other, KE is conserved): m₁v₁ᵢ + m₂v₂ᵢ = m₁v₁f + m₂v₂f

Inelastic collision (objects stick together): m₁v₁ᵢ + m₂v₂ᵢ = (m₁ + m₂)v_f

Topic 5: Electricity Basics

Coulomb's Law

F = kq₁q₂/r² (k = 8.99 × 10⁹ N·m²/C²)

Electric Field

E = F/q = kQ/r²

Ohm's Law

V = IR (Voltage = Current × Resistance)

Power

P = IV = I²R = V²/R

Series Circuits

• Same current through all components
• Voltages add up: V_total = V₁ + V₂ + V₃
• Resistances add: R_total = R₁ + R₂ + R₃

Parallel Circuits

• Same voltage across all branches
• Currents add up: I_total = I₁ + I₂ + I₃
• Resistances: 1/R_total = 1/R₁ + 1/R₂ + 1/R₃

Essential Problem-Solving Tips

1. Units Are Everything

Always include units in your calculations. If your answer is supposed to be in meters and you get seconds, you've made an error. Unit analysis can also help you figure out which equation to use.

2. Sig Figs and Rounding

Don't round until the very end. Rounding intermediate steps introduces errors that compound.

3. Estimate First

Before solving, estimate what a reasonable answer might be. If a car goes from 0 to 60 mph, you should expect an answer around 27 m/s. If you get 2,700 m/s, you've made an error.

4. Check Dimensional Analysis

Every term in an equation must have the same units. If you're adding meters to seconds, something went wrong.

5. Sign Conventions Matter

Choose a direction as positive and be consistent. Most common: right and up are positive, left and down are negative.

6. Draw. Everything.

Free-body diagrams, velocity vectors, circuit diagrams, energy bar charts. Physics is visual, and drawing clarifies your thinking.

Using AI for Physics

Physics is a great subject for AI-assisted learning. Here's why:

• Problems have definite answers, so AI can verify your work
• Step-by-step solutions show exactly where you went wrong
• AI can explain concepts with different analogies until one clicks
• Gradily can walk through problems showing each step's reasoning

Best practice: Try every problem yourself first. When you're stuck, ask AI for a hint (not the full solution). After getting the answer, try a similar problem on your own to verify you actually understood the method.

Physics rewards persistence. The first problem of each type is always the hardest. By the third or fourth similar problem, the approach becomes automatic.

Keep your formula sheet handy, draw your diagrams, check your units, and practice more than you think you need to. Physics isn't about being smart — it's about being systematic.