Acceleration and the v–t graph

"This car does 0 to 100 in 4 seconds." That sentence is an acceleration — a change in velocity, divided by the time it took. It's the second (and last) rate you need for all of kinematics.

Acceleration: the rate velocity changes

$a = \frac{\Delta v}{\Delta t}$

Units tell the story: velocity is meters per second, so acceleration is meters per second per second — m/s². An acceleration of 2 m/s² means the velocity grows by 2 m/s with every tick of the clock.

Two subtleties that trip everyone up:

• Acceleration doesn't mean "speeding up." It means velocity is changing. Slowing down is acceleration (opposite to the motion). So is turning at constant speed — the direction of velocity is changing.
• Negative ≠ slowing down. A negative acceleration speeds you up if you're already moving in the negative direction. The sign is a direction, not a verdict.

The v–t graph, read twice

Put velocity on the vertical axis and the same slope-reading skill from last lesson climbs one level:

• Slope of v–t = acceleration. A straight line means constant acceleration; flat means constant velocity.
• Area under v–t = displacement. Each thin slice of area is (velocity) × (a little time) = a little displacement; stack the slices and the total area is how far you went.

That second reading is the powerful one — it works even when the velocity curve is complicated, and it's the seed of integral calculus.

Reading a motion story

A subway ride between stations is one picture: velocity ramps up (constant positive slope), holds flat (zero acceleration, the cruise), then ramps down to zero (negative slope, the brake). The area under the whole shape — a trapezoid — is the distance between stations.

Why this matters

Slope and area of the v–t graph are the whole toolkit. In the next lesson we point them at the most important special case — constant acceleration — and the famous kinematic equations fall out as pictures.