Motion in One Dimension and Motion Graphs

Key quantities

Quantity	Meaning	Vector or scalar
position x	location relative to an origin	vector
displacement Δx	change in position, x_f − x_i	vector
distance	total path length	scalar
velocity v	rate of change of position, Δx/Δt	vector
speed	magnitude of velocity	scalar
acceleration a	rate of change of velocity, Δv/Δt	vector

A car that drives 5 km east and 5 km back has distance 10 km but displacement 0.

Position-time graphs

• The slope is the velocity.
• A straight line means constant velocity.
• A curve means the velocity is changing (acceleration).

Velocity-time graphs

• The slope is the acceleration.
• The area between the graph and the time axis is the displacement (area below the axis counts negative).

Example. A velocity-time graph rises from 0 to 12 m/s in 4 s. Acceleration = 12/4 = 3 m/s²; displacement = (1/2)(4)(12) = 24 m.

Acceleration and speeding up

An object speeds up when velocity and acceleration point in the same direction, and slows down when they point in opposite directions. Negative acceleration does not always mean slowing down.

Average versus instantaneous

• Average velocity = total displacement ÷ total time.
• Instantaneous velocity = slope of the tangent on a position-time graph.

Quick summary

• Displacement is a vector; distance is not.
• x-t slope = velocity; v-t slope = acceleration; v-t area = displacement.
• Speeding up when v and a have the same sign.