Acceleration is not about how fast you are going. It is about how fast your velocity is changing. Get the small word "change" right, and the equation and the velocity-time graph fall into place together.
The Key Idea
Acceleration is the change in velocity per unit time: a = Δv ÷ t = (v − u) ÷ t, measured in m/s². On a velocity-time graph it is the gradient of the line. A negative value means deceleration (slowing down).
SECTION 01
Acceleration is a gradient.
Drag the start and end points of the line on the graph below. The line's steepness is the acceleration. Notice the trap built into the readout: dividing the final velocity by time (v ÷ t) gives the wrong answer whenever the object did not start from rest.
SECTION 02
Speeding up, slowing down.
Acceleration is positive when the object speeds up (v greater than u).
Deceleration is a negative acceleration: the object slows down, so v is less than u and Δv is negative.
On the graph, an upward slope is acceleration, a downward slope is deceleration, and a flat line is constant velocity (zero acceleration).
The single most common acceleration error
Acceleration is Δv ÷ t, not v ÷ t. If a car speeds up from 4 m/s to 20 m/s in 8 s, the acceleration is (20 − 4) ÷ 8 = 2.0 m/s², not 20 ÷ 8. Only when the object starts from rest (u = 0) do the two happen to agree. Always subtract the starting velocity first.
Worked Example
A train slows from 30 m/s to 18 m/s in 6.0 s. Calculate its acceleration and state what the sign tells you.
Step 1 · Change in velocityΔv = v − u = 18 − 30 = −12 m/s
Step 2 · Divide by timea = Δv ÷ t = −12 ÷ 6.0 = −2.0 m/s²
Step 3 · Interpret the sign
The acceleration is −2.0 m/s². The negative sign shows the train is decelerating (slowing down).