INTRODUCTION: From looking at a position versus time graph (Xt graph), we should be able in principle to: 1} What do the velocity versus time and the acceleration versus time graph look like? 2} How does the object move? Let's compare two \"segments\" of a possibly longer X-t graph (see Figures below): In Figure 1, the Xt graph shows an increasing linear.function. In language more suitable for physical interpretation, we can say that the position of the object becomes (->change, so the object moves) more positive (>so the object moves in the +X-direction) in a gradual manner (-> a gradual change in position means a constant rate-ofchange in position, hence a constant velocity, and, obviously, a zero acceleration). So we first say what we see: the position becomes more positive gradually, and then explain what that means for the moving object: the object moves in the +Xdirection with constant speed (speed is equally correct than velocity since we already specified the direction of motion to be the +Xdirection). The fact that the velocity is constant is easily appreciated if you know calcuius (the derivative of a linear function is a constant), but it is nicer from a physics perSpective to be able to appreciate that conclusion more directly and visually from the fact that, for a small At near the beginning of the time interval [1}, tf] and a small At near the end of the time interval [tg tf], the matching displacements Ax are the same. In other words, the object covers the same amount of terrain x in the same amount of time At throughout the whole interval of observation [t,-, tr], hence the object maintains a constant speed throughout. In Figure 2,the Xt graph shows an increasing function, but not a linear one. In language more suitabie for physical interpretation, we can say that the position of the object becomes {->change, so the object moves) more positive (->so the object moves in the +Xdirection) in a non-gradual manner (> a non gradual change in position means a constant ratewof-change in position, hence a changing velocity and a non-zero acceleration). Perhaps the best way-to say what we see is that the position becomes more positive at a faster and faster rate (the graph shows "explosive" behavior). The spontaneous conclusion as to what that means for the moving object is that the object moves in the +X-direction while speeding up. Of course, knowing calculus makes it quickly clear that the derivative of the position function yields a positive velocity function that itself becomes more positive (the velocity becomes more positive or the object speeds up in the +X-direction). in the particular situation where the position function is given to be quadratic, the velocity function will be linear. Again, it is nicer from a physics perspective to be able to appreciate that conclusion more directiy and visually from the fact that, for a small zit near the beginning of the time interval [tg tf] and a small At near the end of the time interval [1.3, tf], the matching displacements Ax are not the same anymore: where the graph is steeper, the displacement is larger. In other words, the object covers more terrain [Ix in the same amount oftime At near the end of the whole interval of observation [tg ti] than in the beginning, hence the object speeds up. ASSIGNMENT: Try to analyze the following graphs in the same way as the preceding examples, that is: 1) state what you see in the position versus time graph (you may assume that all curved pieces are parabolicll} 2) state what the implication is for the velocity versus time graph as well as for the acceleration versus time graph {no need to bring in calculus knowledge, just use good technical words) 3) draw onto the given graph the displacements x for a small time interval Atn'ear the beginning and near the end of [ti, tr] 4) nally, describe in a short English sentence how the object is moving during [tb tf]