Figure 2-50 shows the position vs. time graph for two bicycles, A and B. (a) Identify any instant at which the two bicycles have the same velocity. (b) Which bicycle has the larger acceleration? (c) At which instant(s) are the bicycles passing each other? Which bicycle is passing the other? (d) Which bicycle has the largest instantaneous velocity? (e) Which bicycle has the larger average velocity?

In problems related to distance vs. time graphs, the instantaneous slope plays a crucial role in determining the velocity.

For example, when the slopes of any two particular objects are similar, it means that their velocities will be equal.

As observed from the graph, both bicycles will have identical velocities at any instant if the instantaneous slopes of their distance vs. time graph are similar. This occurs at time \({t_1}\), which is marked on the graph below.

In the given graph, it can be seen that bicycle B has no acceleration because it has a constant slope. On the other hand, bicycle A has positive acceleration because the graph is concave upward.

Thus, bicycle A will have more acceleration.

In the graph, when the two graphs will cross each other, the bicycles will pass each other. This is because A and B will have the same position at that time. The graph with the steepest slope is the faster bicycle. At the first crossing, bicycle B is crossing bicycle A, and at the second crossing bicycle A is passing bicycle B.

From the graph mentioned in part (a), it can be said that bicycle B will have the largest instantaneous velocity until time \({t_1}\), where the graphs have identical slopes. Bicycle A has the largest instantaneous center throughout the time after \({t_1}\). At the latest time shown in the graph, bicycle A has the largest instantaneous velocity.

When the starting point of a graph for any bicycle, A or B, is attached to the ending point with a straight line, the slope of that line forms the average velocity. Also, for that average line, both bicycles will have similar average velocities.