1D Motion Essentials: Position, Velocity, and Acceleration

Position, Distance, and Displacement

Position
: 1D X/Y, units in meters.

Distance traveled (L)
: total path length, always {\geq}0.

Displacement
: vector from start to finish. If returning to start, L > 0, but displacement is 0.

Velocity and Speed

Velocity units
: meters per second.

Instantaneous velocity
: how fast something moves and its direction at a specific moment.

Average velocity
: change in position over a time interval.

Average speed
: total path length traveled over a time interval.

Direction of motion
: same as instantaneous velocity. Positive velocity means moving right/up; negative velocity means moving left/down. Direction changes when velocity crosses zero.

Acceleration

Acceleration units
: meters per second squared.

Instantaneous acceleration
: how fast velocity is changing at a specific moment.

Average acceleration
: change in velocity over a time interval.

Motion
: Speeding up if velocity and acceleration have the same direction (same sign); slowing down if they have opposite directions (opposite signs).

Calculus and Graphical Relationships

Position-time (X-t) graph
: Slope represents velocity.

Velocity-time (Vx-t) graph
: Slope represents acceleration.

Displacement
: Area under the velocity-time graph.

Velocity change
: Area under the acceleration-time graph.

Position
: Integral of velocity over time, plus initial position.

Velocity
: Integral of acceleration over time, plus initial velocity.

Key Concepts

Constant acceleration
: Instantaneous acceleration equals average acceleration.

Average acceleration
: Velocity change over time.

Kinematics Equations (constant acceleration)

• Four equations relate displacement, velocities, acceleration, and time.

• All involve initial velocity; two equations may be needed for two unknowns.

Problem-solving guidance

• Identify knowns and unknowns.

• Select appropriate equations; two might be needed if initial velocity is unknown.

Graphical interpretation (one-dimensional motion)

Acceleration–time (a-t)
: horizontal line.

Velocity–time (v-t)
: straight line (slope = acceleration, intercept = initial velocity).

Position–time (x-t)
: parabola (slope = velocity).

Free fall (gravity-only motion)

• Motion under gravity, neglecting air resistance.

• Gravity magnitude
  : Approximately 9.8\ \mathrm{m/s^2}.

• If positive is upward, gravitational acceleration is downward.

Important conventions and notes

Vertical coordinates
: Positive up means gravity accelerates negatively.

• Ensure consistent initial quantities and signs.