Final answer:
The pair of graphs that represent the same motion will illustrate equal vertical displacement over equal time intervals for both balls, regardless of their horizontal motions. One graph will show a constant horizontal displacement for the ball with initial horizontal velocity, while the other will have no such displacement for the ball at rest horizontally.
Step-by-step explanation:
The question regarding which pair of graphs represent the same motion of an object relates to Physics, and the concept in focus is the independence of vertical and horizontal motions in two-dimensional motion.
According to the information provided, we are looking at a scenario with two balls.
One ball drops from rest while the other has an initial horizontal velocity. However, their vertical velocities and positions are identical over time, which illustrates that the vertical motion is unaffected by whether or not there is horizontal motion. Graphicals representations of such motions will show that the positions of the balls with respect to the vertical axis are the same when plotted against time.
Therefore, the graphs that represent their vertical motion will be the same (consistent vertical displacement over equal time intervals), while the horizontal motion of one ball will be represented as a constant horizontal displacement over time, and for the other ball at rest horizontally, the graph will show no horizontal displacement.