Consider motion in which the velocity grows at a steady rate. That steady rate is what we call acceleration: the change in velocity divided by the time it took.
Written out, it is . A velocity is being divided by a time, so the unit is divided once more by seconds, which is .
Put on a - graph, the points fall on a single straight line. The line rises by while it runs across, so that ratio, the slope of the line, is the acceleration itself. The larger the acceleration, the more steeply the graph climbs.
The dots along the path in the figure are stamped at equal intervals of time. The faster the object moves, the farther it gets in the same interval, so the gaps between the dots widen toward the right.
Set the acceleration to zero for a moment and watch motion at an unchanging velocity. The stamped dots are evenly spaced, and the distance covered is . Speed times time, the easiest calculation there is.
Seen on the - graph, though, the same fact can be said another way. The velocity is a horizontal line, and beneath it, down to the time axis, stands a rectangle. Its height is and its width is , so the area of that rectangle is exactly , the distance covered.
That speed times time comes out as an area is only a restatement of the obvious. But the restatement is a strong one, because the area still works when the velocity changes.
When the velocity keeps changing, there is no telling which value belongs in the of . So we cut the time into short pieces.
Over a piece short enough, the velocity barely changes. The distance covered in that piece can therefore be replaced by the area of a thin rectangle. Build one rectangle per piece and add them all together, and the total is the distance covered.
As the pieces grow finer, the staircase of rectangles draws closer to the shape under the line. The pieces can be made as fine as we like, so we are entitled to say it outright: the distance covered is the area enclosed by the graph and the time axis.
Whether the velocity changes or not, area is displacement. This is the most important rereading in the topic.
The shape waiting at the end of all that chopping is a trapezoid, with and as its two parallel sides and as the distance between them. Its area is .
The trapezoid splits into a rectangle below and a triangle above. The rectangle is the part covered at the initial speed, . The triangle is the part added by accelerating, and since it stands tall across a width of , it comes to . Adding them gives .
The two forms are the same thing. is the average of the initial and final velocities, which is the average velocity. Under constant acceleration that average sits exactly halfway between the two, so the motion may be treated as though it ran at that one speed throughout, and the time simply multiplied in.