AP Physics C: Mechanics — One-Dimensional Kinematics (Unit 1 Study Notes)

Position (x or x(t))

The coordinate location of an object along a chosen axis; written as x(t) to emphasize that position can change with time.

Reference point (origin)

The location defined as x = 0; choosing it sets the coordinate system but does not change the physics of the motion.

Positive direction

The chosen direction along the axis that counts as positive; it determines the sign of position, displacement, velocity, and acceleration.

Displacement (Δx)

The signed change in position: Δx = xf − xi; can be positive, negative, or zero.

Distance traveled

The total path length traveled (always nonnegative); can differ from displacement if direction changes.

Average velocity (v_avg)

Net displacement divided by elapsed time: v_avg = Δx/Δt; depends only on start and end positions, not the path in between.

Instantaneous velocity (v(t))

The time-derivative of position: v(t) = dx/dt; gives the velocity at a specific instant.

Slope of an x–t graph

Represents velocity at that time; a steeper slope means larger speed, and the sign gives direction.

Average acceleration (a_avg)

Change in velocity over change in time: a_avg = Δv/Δt.

Instantaneous acceleration (a(t))

The time-derivative of velocity: a(t) = dv/dt.

Second derivative relationship

Acceleration as the second derivative of position: a(t) = d²x/dt².

Area under a v–t graph

The signed area equals displacement: Δx = ∫ v(t) dt; regions below the time axis contribute negative displacement.

Area under an a–t graph

The signed area equals change in velocity: Δv = ∫ a(t) dt; regions below the axis contribute negative Δv.

Signed area (graph interpretation)

Area counted with sign relative to the axis; positive above the axis and negative below, so areas can cancel.

Constant acceleration

Acceleration a(t) stays the same value over the time interval being analyzed (can be positive or negative, but not changing with time).

Velocity under constant acceleration

For constant a, velocity varies linearly with time: v(t) = v_0 + at.

Position under constant acceleration

For constant a, position follows: x(t) = x0 + v0 t + (1/2)at².

Time-free kinematic equation

For constant a, relates velocity and position without time: v² = v0² + 2a(x − x0).

Average-velocity form (constant a)

For constant acceleration, displacement relates to average of initial and final velocity: x − x0 = (1/2)(v + v0)t.

Free-fall sign convention

The magnitude of gravity is g ≈ 9.8 m/s², but its sign depends on your chosen positive direction (e.g., up positive ⇒ a = −g).

Turning point misconception

At maximum height (v = 0) in vertical motion, acceleration is not zero; it remains a = −g (ignoring air resistance).

Speeding up vs slowing down (sign test)

If v and a have the same sign, speed increases; if v and a have opposite signs, speed decreases.

Non-constant acceleration (calculus approach)

When acceleration varies, constant-a formulas don’t apply; use v(t)=v(t0)+∫a(t)dt and x(t)=x(t0)+∫v(t)dt.

Chain-rule identity (a as a function of x)

If a depends on position, use a = v(dv/dx) to connect acceleration, velocity, and position without solving for time.

Initial conditions (integration constant)

Values like v(t0) and x(t0) needed when integrating; they determine the constants so the resulting v(t) and x(t) are unique.