# Velocity time graph analysis: what does a concave downward $v$-$t$ curve mean?

This is a screenshot from the lecture about the analysis of various velocity-time graphs I was watching.

I understand that

the concavity of velocity-time graph will tell about the increasing or decreasing nature of acceleration.

Therefore, this graph must have decreasing acceleration with time since the curve is clearly concave downwards.

Then, why it is written "increasing acceleration" ?

## 2 Answers

Instead of concave or convex it's easier to note the change in the gradient of the graph to see how the acceleration changes. Here the acceleration ( the gradient of the curve) has a low negative ( suppose -1) then it becomes steeper with time so it has a negative value of larger magnitude ( suppose -4). Thus the magnitude of the acceleration is increasing. Thus you are right as the absolute acceleration is ofcourse decreasing. Hope I could help.

• Why does the "concavity" reasoning fails? Jun 8, 2017 at 13:16
• Does not fail, edited. Jun 8, 2017 at 13:18
• So, the concavity will only tell about the "magnitude" of increase or decrease of acceleration? Jun 8, 2017 at 13:24
• Yes, correct and the direction if its positive or negative acceleration can be found by seeing the slope of curve. Jun 8, 2017 at 13:25
• Please verify answer if you're satisfied with it. Jun 8, 2017 at 13:26

Velocity is related to acceleration by: $$\vec{a}=\frac{d\vec{v}}{dt}$$

Think in terms of the slope of the velocity-time graph.

To study acceleration from this graph, you have to see how the slope is varying w.r.t each point.

(This graph does have decreasing acceleration. Only the magnitude increases.)

• Slope is negative since theta is obtuse. Slope only tells us about the increasing or decreasing nature of "velocity", not acceleration. What am I missing? Jun 8, 2017 at 13:18
• The slope of this curve is the acceleration. The slope becomes more negative as you go along the graph. Jun 8, 2017 at 13:20
• The increasing or decreasing nature of velocity IS acceleration! Jun 8, 2017 at 13:20