The difference between "size only" and "size with direction"
The Big Idea
Some quantities — like temperature or mass — are fully described by just a single number. These are scalars. Other quantities — like velocity or displacement — also need a direction to be meaningful. These are vectors. Knowing which is which is foundational to all of physics.
Key Definitions
Scalar
Magnitude only. Examples: distance, speed, time, mass.
Vector
Magnitude AND direction. Examples: position, displacement, velocity, acceleration.
Magnitude
The size of a quantity, ignoring direction. Always positive.
Vector Sum (1D)
Adding vectors along the same line — opposite directions get opposite signs.
Example
Walking around the block
You walk 5 m east, then 3 m west. Your distance (a scalar) is 5 + 3 = 8 m.
Your displacement (a vector) is +5 + (−3) = +2 m east. The opposite direction got the negative sign.
Topic 1.2
Displacement, Velocity, and Acceleration
The three quantities that describe motion
The Big Idea
These three quantities form the entire vocabulary of motion. Velocity tells you how fast position is changing; acceleration tells you how fast velocity is changing. Each is a vector — direction matters. The "object model" lets us treat any object as a single point so we can focus on its motion.
Key Definitions
Displacement (Δx)
Change in position. Vector. x = x_final − x_initial.
Average velocity
v_avg = Δx / Δt. Same direction as displacement.
Instantaneous velocity
Velocity at one instant — the slope of a tangent on the x-t graph.
Average acceleration
a_avg = Δv / Δt. An object accelerates if speed OR direction changes.
Common Pitfall
An object can have zero velocity but non-zero acceleration. A ball thrown straight up has v = 0 at its peak, but gravity is still pulling on it (a = g downward). Don't confuse "not moving right now" with "not accelerating."
Example
A sprinter speeds up
A sprinter goes from 0 m/s to 10 m/s in 2 seconds. The average acceleration is:
a_avg = Δv / Δt = (10 − 0) / 2 = 5 m/s² in the direction of motion.
Topic 1.3
Representing Motion
Graphs, equations, and motion diagrams
The Big Idea
The same motion can be described with motion diagrams, equations, OR graphs. The kinematic equations work for constant acceleration. The connections between graphs are critical: slope of one graph gives the next quantity; area under one graph gives the previous quantity.
The Kinematic Equations
v = v₀ + at — connects velocity, acceleration, and time
x = x₀ + v₀t + ½at² — connects position, velocity, acceleration, and time
v² = v₀² + 2a(x − x₀) — when you don't have (or need) time
These ONLY work for constant acceleration. Near Earth's surface, gravity gives g ≈ 10 m/s² downward.
Going UP the chain (a → v → x): take area. Going DOWN (x → v → a): take slope.
Example
Dropped from rest
A rock is dropped from a 20 m cliff. How long until it hits the ground?
Use x = x₀ + v₀t + ½at². Set x = 0 (ground), x₀ = 20 m, v₀ = 0, a = −10 m/s². Solve: t = √(20·2 / 10) = 2 s.
Topic 1.4
Reference Frames and Relative Motion
Different observers measure different velocities
The Big Idea
Motion is always measured from somewhere — the observer's reference frame. Different observers can measure different velocities for the same object, but if both frames are inertial (not accelerating), they'll always agree on the acceleration. For AP Physics 1, relative motion problems are limited to one dimension.
Key Definitions
Reference frame
The viewpoint from which motion is measured.
Inertial frame
A frame that isn't accelerating. Newton's first law holds.
Relative velocity
Velocity measured from one specific frame. Add velocities to switch frames.
Big takeaway
All inertial observers agree on acceleration, even if they disagree on velocity.
Example
Walking on a train
You walk forward on a train at 2 m/s (relative to the train). The train moves east at 30 m/s (relative to the ground).
Your velocity relative to the ground = 2 + 30 = 32 m/s east.
If you walked backward at 2 m/s on the train: 30 − 2 = 28 m/s east. Direction matters — add signed velocities.
Common Pitfall
Just because someone in a different reference frame measures a different velocity, doesn't mean the laws of physics are different. Acceleration is the same in all inertial frames — which is why Newton's second law (F = ma) works regardless of who's watching.
Topic 1.5
Vectors and Motion in Two Dimensions
2D motion is just two 1D problems happening at once
The Big Idea
To handle 2D motion, break vectors into perpendicular components (usually x and y). The horizontal and vertical motions are completely independent — they only share time. Projectile motion is the classic case: zero horizontal acceleration; vertical motion is free fall (a = g downward).