Motion Basics

A jogger runs one full lap around a 400 m track and finishes exactly where she started. She traveled 400 meters of distance — but her displacement is zero. She might have averaged a steady 5 m/s in speed, but her average velocity over the whole lap is also zero.

Wait, what?

This isn’t a word game. Distance, displacement, speed, and velocity are four different physical quantities, and the MCAT writes traps that look identical until you notice which word the question used. This section is where you nail those distinctions down for good.

Position and Displacement

Position ($x$) is where something is, measured from a chosen origin. “The ball is at $x = 5$ m” only makes sense if you’ve already decided where $x = 0$ is.

Displacement ($\Delta x$) is the change in position — how far the object ended up from where it started, plus the direction. It’s a vector.

Distance is the total length of the path traveled. It’s a scalar — always positive, and always greater than or equal to the magnitude of displacement.

| Quantity | Type | Depends on path? | Can be zero for a round trip? |
|----------|------|-------------------|-------------------------------|
| Distance | Scalar | Yes | No (always positive if motion occurred) |
| Displacement | Vector | No | Yes (if start = finish) |

Speed and Velocity

Speed is the rate at which distance changes. Scalar. Always positive.

Velocity is the rate at which displacement changes. Vector. Can be positive, negative, or zero.

The relationship mirrors the one above: speed pairs with distance, velocity pairs with displacement.

Example. A runner finishes a 400 m lap in 80 seconds.

• Average speed = $400 / 80 = 5$ m/s.
• Average velocity = $0 / 80 = 0$ m/s (because displacement is zero).

Same runner. Same 80 seconds. Same physical motion. Two completely different “average” values, depending on which word the question uses. Read carefully.

Average vs. Instantaneous

There’s another pairing that trips students up: average vs. instantaneous.

• Average velocity is your displacement over a chunk of time. It tells you the overall trend.
• Instantaneous velocity is your velocity at one specific moment — what the speedometer reads right now. On a position-vs-time graph, it’s the slope of the line tangent to the curve at that point.

Imagine a 4-hour road trip where you average 60 mph. That doesn’t mean you went 60 mph the whole time — you probably went 75 on the highway, 30 in town, and 0 at a rest stop. Each of those is an instantaneous velocity at some moment. The average just smooths them all together.

Acceleration

Acceleration is the rate at which velocity changes. Vector.

The biggest misconception about acceleration: students hear “acceleration” and think “speeding up.” That’s only one type. Acceleration just means velocity is changing somehow — speeding up, slowing down, or changing direction. All three count.

• Speeding up in your direction of motion → acceleration in the same direction as velocity.
• Slowing down → acceleration opposite to velocity (often called deceleration; physicists call it negative acceleration).
• Constant velocity (steady speed and steady direction) → zero acceleration, even at 100 mph.
• Turning at constant speed → still accelerating, because direction is changing.

Graphs of Motion

Use the simulation below to see how position, velocity, and acceleration graphs connect. Pick a motion type and watch all three graphs draw simultaneously. The key insight: each graph is the slope of the one above it.

| Graph | Slope gives you… | Area under curve gives you… |
|-------|---------------------|-------------------------------|
| Position vs. time | Velocity | — |
| Velocity vs. time | Acceleration | Displacement |
| Acceleration vs. time | — | Change in velocity |

This “slopes go down, areas go up” relationship is the conceptual backbone of every kinematics graph problem. Memorize it as a mental staircase: position → velocity → acceleration each step is a slope.

Sign Conventions

Always pin down a positive direction before doing math. Standard convention: rightward and upward are positive.

Once that’s set, the relationship between velocity and acceleration tells you whether the object is speeding up or slowing down:

• Positive $v$ + positive $a$ → speeding up (in the positive direction).
• Positive $v$ + negative $a$ → slowing down.
• Negative $v$ + negative $a$ → speeding up (in the negative direction).
• Negative $v$ + positive $a$ → slowing down.

The shortcut: if $v$ and $a$ have the same sign, the object speeds up. If they have opposite signs, the object slows down. The actual signs (positive or negative) just tell you which direction the object is moving and which way the force is pushing.