The acceleration-time graph
The acceleration-time graph of any object traveling with a constant velocity is the same. This is true regardless of the velocity of the object. An airplane flying at a constant 600 mph (270 m/s), a sloth walking with a constant speed 1 mph (0.4 m/s), and a couch potato lying motionless in front of the TV for hours will all have the same acceleration-time graphs — a horizontal line collinear with the horizontal axis. That's because the velocity of each of these objects is constant. They're not accelerating. Their accelerations are zero. As with velocity-time graphs, the important thing to remember is that the height above the horizontal axis doesn't correspond to position or velocity, it corresponds to acceleration.
If you trip and fall on your way to school, your acceleration towards the ground is greater than you'd experience in all but a few high performance cars with the "pedal to the metal". Acceleration and velocity are different quantities. Going fast does not imply accelerating quickly. The two quantities are independent of one another. A large acceleration corresponds to a rapid change in velocity, but it tells you nothing about the values of the velocity itself.
When acceleration is constant, the acceleration-time curve is a horizontal line. The rate of change of acceleration with time is a meaningless quantity so the slope of the curve on this graph is also meaningless. Acceleration need not be constant, but the time rate of change of this number has no name. On the surface, the only information one can glean from an acceleration-time graph is the acceleration at any given time.
On an acceleration-time graph …
slope is meaningless.
the "y" intercept equals the initial acceleration.
when two curves coincide, the two objects have the same acceleration at that time.
an object undergoing constant acceleration traces a horizontal line.
zero slope implies motion with constant acceleration.
Acceleration is the rate of change of velocity with time. Transforming a velocity-time graph to an acceleration-time graph means calculating the slope of a line tangent to the curve at any point. (In calculus, this is called finding the derivative.) The reverse process entails calculating the cumulative area under the curve. (In calculus, this is called finding the integral.) This number is then the change of value on a velocity-time graph.
Given an initial velocity of zero (and assuming that down is positive), the final velocity of the person falling in the graph to the right is …
Δv = aΔtΔv = (9.8 m/s2)(1.0 s)Δv = 9.8 m/s ≈ 20 mph
and the final velocity of the accelerating car is …
Δv = aΔtΔv = (5.0 m/s2)(6.0 s)Δv = 30 m/s ≈ 60 mph
On an acceleration-time graph …
the area under the curve equals the change in velocity.
There are more things one can say about acceleration-time graphs, but they are trivial for the most part.
Source:http://physics.info/motion-graphs/