Question Video: Finding the Time Intervals in Which the Acceleration of a Particle Is Negative Using a Velocity–Time Graph Mathematics

The figure shows a velocity–time graph for a particle moving in a straight line. When is the acceleration of the particle negative?

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Video Transcript

The figure shows a velocity–time graph for a particle moving in a straight line. When is the acceleration of the particle negative?

In order to answer this question, let’s recall what we know about the relationship between velocity and acceleration. Acceleration is defined as the rate of change of velocity of an object. On a velocity–time graph, we interpret the acceleration as being equal to the slope of the curve. And so what we’re really asking ourselves here is, when is the slope of our velocity–time graph negative?

Well, we can see that this occurs in several places. If we look at this line segment here, it has a negative slope; it slopes downwards. We also have this line segment here. And there’s one further line segment that slopes downwards. It’s this one. The slope of the remaining line segments is either zero, that’s here and here, or positive, that’s here and here.

We can, therefore, say that since the slope is negative on these three line segments, the acceleration itself is negative. This occurs over the first four seconds, so from 𝑡 equals four to 𝑡 equals zero. And then it occurs again from 𝑡 equals nine to 𝑡 equals 11. By considering the slope of the velocity–time graph, we see its acceleration is negative from 𝑡 equals zero to 𝑡 equals four and from 𝑡 equals nine to 𝑡 equals 11.

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