One-Dimensional Motion in AP Physics

What Is One-Dimensional Motion?

One-dimensional motion is movement that happens along a single axis. That's it. A car driving down a straight road. A ball thrown straight up. A person walking forward. All of these are examples of 1D motion.

In AP Physics, you'll spend a significant chunk of time mastering this topic because it forms the foundation for everything that comes after. Vectors, forces, energy — none of it makes sense if you can't handle motion in a straight line first.

The key assumption here is that objects move along the x-axis (horizontal) or y-axis (vertical), but never both at the same time in a single problem.

The Core Quantities You Must Know

Before you can solve any 1D motion problem, you need to understand what you're actually measuring. These five quantities are the language of kinematics.

Displacement (Δx)

Displacement is the change in position from start to finish. It's a vector, meaning it has direction. Unlike distance, displacement doesn't care about the path you took — only where you ended up relative to where you started.

Formula: Δx = x₂ - x₁

Velocity (v)

Velocity is displacement divided by time. It tells you how fast something is moving and in which direction.

Average velocity: v_avg = Δx / Δt

Instantaneous velocity is what your speedometer reads at any specific moment — the limit of average velocity as the time interval shrinks to zero.

Acceleration (a)

Acceleration is the rate of change of velocity. When velocity changes — whether speeding up, slowing down, or changing direction — you're dealing with acceleration.

Formula: a_avg = Δv / Δt

⚠️ Critical trap: Many students assume acceleration always means speeding up. It doesn't. Acceleration simply means velocity is changing. A car braking hard has high acceleration even though it's slowing down.

Time (t)

Time is the independent variable in most 1D problems. You measure it in seconds. Everything else gets expressed in terms of t or Δt.

Initial vs. Final Conditions

Physics problems almost always give you initial conditions (usually at t = 0) and ask you to find final conditions. Get in the habit of identifying:

The Four Kinematic Equations

These are your bread and butter. Pick the equation that contains three known variables and solve for the fourth. That's the entire strategy for most 1D motion problems.

Equation What It Contains Best Used When
v = v₀ + at v, v₀, a, t You know v₀, a, t and need v
x = x₀ + v₀t + ½at² x, x₀, v₀, a, t You know v₀, a, t and need position
v² = v₀² + 2a(x - x₀) v, v₀, a, Δx Time is unknown — no t needed
x - x₀ = ½(v₀ + v)t x, x₀, v₀, v, t You know v₀, v, t and need displacement

⚠️ These equations only work when acceleration is constant. If acceleration is changing, you need calculus — which is exactly what you'll learn later in the course.

Free Fall: The Special Case

When an object falls near Earth's surface with no air resistance, its acceleration is constant at g = 9.8 m/s² (or 10 m/s² on the AP exam for simpler math).

Direction matters here. By convention:

Pick one direction and stick with it throughout the entire problem. Mixing signs is the fastest way to get the wrong answer.

Example: You throw a ball upward at 20 m/s. It rises, stops, and falls back down. At its highest point, velocity = 0 — but acceleration is still -9.8 m/s². The ball isn't weightless or suspended in midair. It's still accelerating downward the entire time.

Reading Motion Graphs

Graphs show up constantly on the AP exam. You need to be fluent in reading them.

Position vs. Time Graphs

Velocity vs. Time Graphs

Acceleration vs. Time Graphs

How To Solve Any 1D Motion Problem

Follow this sequence every time. No exceptions.

Step 1: Read the problem twice

First read: Understand the scenario. Second read: Identify what you're solving for.

Step 2: List your known variables

Write down everything given: v₀, v, a, Δx, t. Circle what you're solving for.

Step 3: Choose your coordinate system

Pick a positive direction. Usually forward or upward. Write it down. This keeps your signs consistent.

Step 4: Select the right kinematic equation

Match an equation to your three knowns and one unknown. If time isn't given or needed, use the equation without t in it.

Step 5: Plug in the numbers

Include units. Check that they're consistent (convert km/h to m/s if needed). Don't forget to carry signs through — negative velocity and negative acceleration are valid inputs.

Step 6: Solve algebraically first, then calculate

Solve for the unknown in symbols first. Then substitute numbers. This prevents arithmetic errors and shows your work if partial credit matters.

Step 7: Check your answer

Does the sign make sense? Is the magnitude reasonable? If you throw a ball up at 15 m/s, it won't reach a height of 500 meters. Trust your gut.

Common Mistakes That Cost Points

Units and Conversions

The AP exam uses SI units. Get comfortable converting between them:

Comparing 1D Motion Variables

Quantity Symbol Vector? Can Be Negative? SI Unit
Displacement Δx Yes Yes meter (m)
Velocity v Yes Yes m/s
Acceleration a Yes Yes m/s²
Time t No No second (s)

Where 1D Motion Leads

Once you master 1D motion, you extend it to two dimensions by treating horizontal and vertical components separately. Projectile motion is just 1D motion with gravity acting on the vertical component. Circular motion applies these same principles with constant acceleration directed toward the center.

The physics doesn't change — only the math gets more complex. Master the fundamentals now, and 2D problems become tedious rather than impossible.