Question

Why does a curve on a distance-time graph indicate acceleration?

Answer 1

A curve on a distance-time graph indicates distance with respect to time.

For someone good in interpreting curves, the slope of distance/time at each point on the curve represents the velocity.

For someone still better in reading curves, the second derivative of the curve at any point indicates acceleration. Second derivative means the rate of variation of velocity with respect to time.

So the quick answer is a curve on a distance time graph does NOT directly indicate acceleration. See example in attached diagram. If you can "see" the constant acceleration indicated in the bottom (distance-time) graph, then yes, you can find acceleration from a distance-time graph. Otherwise, the answer is still no.