Physics: Kinematics Key Concepts and Graph Interpretations

displacement

Vector from initial to final position (units: meters, m). Direction matters.

distance (vs. displacement)

Total path length traveled (scalar). Distance ≥ |displacement|.

SI unit for acceleration

meters per second squared (m/s²).

Average velocity formula and units

𝑣ˉ=Δ𝑥/Δ𝑡 (m/s).

Average acceleration formula and units

𝑎ˉ=Δ𝑣/Δ𝑡 (m/s²).

Instantaneous velocity

Slope (derivative) of position-time at a point: 𝑣=𝑑𝑥/𝑑𝑡.

Instantaneous acceleration

Slope (derivative) of velocity-time at a point: 𝑎=𝑑𝑣/𝑑𝑡.

Key equation for constant acceleration

𝑣²=𝑣₀²+2𝑎(𝑥−𝑥₀).

Position-time graph with zero acceleration

A straight line (constant slope = constant velocity).

Velocity-time graph with constant nonzero velocity

A horizontal (flat) line at that velocity value.

Slope of a position-time graph

Velocity (positive slope → moving in + direction; negative slope → moving in − direction).

Slope of a velocity-time graph

Acceleration.

Area under a velocity-time graph

Displacement (not distance unless velocity is always nonnegative).

Area under an acceleration-time graph

Change in velocity, Δ𝑣.

Position-time graph: parabola opening upward

Indicates constant positive acceleration (position varying as 𝑥(𝑡)=𝑥₀+𝑣₀𝑡+1/2𝑎𝑡² with 𝑎>0).

Reversing direction from graphs

On v-t: velocity crosses zero (sign change). On x-t: slope changes sign (from + to − or vice versa).

Speed from a v-t graph when velocity goes negative

Take area using absolute value for distance; for displacement, keep sign.

Relationship of v-t straight line with negative slope

Indicates constant negative acceleration (speed decreasing if v positive, or speed increasing in negative direction if v negative).