AP Physics 1 Kinematics: Building One-Dimensional Motion from First Principles

One-dimensional motion

A model of motion where position is described along a single axis (e.g., x-axis), with direction represented by positive/negative signs.

Coordinate system (1D)

Your chosen origin (x = 0) and positive direction for describing motion along one line.

Sign convention

A stated choice of which direction is positive (e.g., “right is positive”); must be kept consistent throughout a problem.

Scalar

A quantity with magnitude only and no direction (e.g., time, distance, speed).

Vector (in 1D)

A quantity with magnitude and direction, where direction is indicated by the sign (+/−) (e.g., displacement, velocity, acceleration).

Position (x)

An object’s location along the chosen axis relative to the origin; can be positive, negative, or zero.

Displacement (Δx)

The signed change in position, defined as Δx = xf − xi; a 1D vector.

Distance traveled

Total path length covered; always nonnegative and does not cancel when direction changes.

Speed

How fast an object moves with no direction; average speed = (distance traveled)/(time) and cannot be negative.

Time interval (Δt)

Elapsed time defined as Δt = tf − ti; typically positive in physics problems.

Average velocity (v_avg)

Displacement per time: v_avg = Δx/Δt; depends only on start and end positions, not the path.

Instantaneous velocity

Velocity at a specific moment in time; conceptually the slope of the position-time graph at a point.

Average acceleration (a_avg)

Change in velocity per time: aavg = Δv/Δt, where Δv = vf − v_i.

Magnitude of a 1D vector

The absolute value of the signed quantity (e.g., |−7 m/s| = 7 m/s).

Constant acceleration

Acceleration that remains the same value over the entire time interval; enables the standard 1D kinematics equations.

Kinematics equation: v = v₀ + at

Constant-acceleration relationship connecting final velocity, initial velocity, acceleration, and time.

Kinematics equation: x = x₀ + v₀t + (1/2)at²

Constant-acceleration relationship giving position after time t from initial position, initial velocity, and acceleration.

Kinematics equation: v² = v₀² + 2a(x − x₀)

Constant-acceleration relationship that connects velocity and position without using time.

Kinematics equation: Δx = ((v₀ + v)/2)t

Under constant acceleration, displacement equals average of initial and final velocities times time.

Speeding up vs slowing down (sign rule)

In 1D, if velocity and acceleration have the same sign, speed increases; if they have opposite signs, speed decreases.

Motion diagram

A diagram of dots at equal time intervals; dot spacing shows speed changes and velocity arrows can show direction and relative speed.

Position-time graph (x vs t)

Graph of position versus time where the slope represents velocity; steeper slope means greater speed.

Velocity-time graph (v vs t)

Graph of velocity versus time where slope represents acceleration and area under the curve represents displacement.

Acceleration-time graph (a vs t)

Graph of acceleration versus time where the signed area under the curve represents change in velocity (Δv).

Turning point (change of direction)

The moment an object reverses direction; in 1D this occurs when velocity v = 0 (not necessarily when x = 0).