Figure 2-42 shows the velocity of a train as a function of time. (a) At what time was its velocity greatest? (b) During what periods, if any, was the velocity constant? (c) During what periods, if any, was the acceleration constant? (d) When was the magnitude of the acceleration greatest?

The graph plotted between the velocity of an object and time is referred to as the velocity-time graph. You can measure acceleration and displacement with the help of a velocity-time graph.

For example, the slope of the velocity-time graph at any point will give you the object's acceleration at that point. However, the area enclosed by the graph with the time axis between a certain time period will give you the object's displacement in that period.

The train's velocity was the greatest at the time that corresponds to the peak of the velocity-time graph, i.e., the highest point of the graph. The peak of this velocity- occurred when the time was just near 50 seconds. You can approximate it to be at 48 seconds.

Thus, you can say that the train's velocity was the greatest at time t = 48 s.

The train’s velocity was constant in that time period corresponding to which the velocity-time graph was horizontal, i.e., the time period in which the value of the train's velocity remained the same.

It can be seen from the graph that the velocity of the train is zero from the time just near 90 s to the time just before 110 s. Thus, you can say that the velocity was constant approximately between time t = 88 s to t = 108 s.

So, the train's velocity was constant between the time period starting from 88 s to 108 s.

The slope of the velocity-time graph gives the acceleration of the object. A straight line in the velocity-time graph represents that the object's velocity increases or decreases at a constant rate with respect to time. Thus, you can say that the slope of the graph, i.e., acceleration remains constant when a part of the velocity-time graph is straight.

It can be seen from the graph that the velocity of the train was increasing at a constant rate between the time period 0 s to 40 s and decreasing at a constant rate between 60 s to 80 s.

Thus, the train's acceleration was constant between the time period of 0 s to 40 s and between 60 s to 80 s.

You already know that the slope of the velocity-time graph determines the acceleration. The acceleration of an object is the greatest when the slope of the graph is steep, i.e., when there is a large change in velocity for a small change in time.

It can be seen from the graph that the slope of the graph is the steepest between the time period 60 s to 80 s. Although the velocity is decreasing, i.e., acceleration is negative in this time period, you only have to see the magnitude of acceleration.

Therefore, the acceleration of the train was the greatest between the time period 60 s to 80 s.