An object is given a short horizontal push that sets it in motion along a smooth horizontal surface. When the object reaches the end of the surface, it undergoes projectile motion from an initial position to a final position as shown in the diagram. Which of the graphs (a), (b), (c), and (d) shows the changes in the horizontal distance moved by the object between its initial and final positions?
To get started, let’s look at our original diagram that shows our object being pushed by a force 𝐹 across a smooth horizontal surface. Once the object reaches the edge of the surface, it begins projectile motion. In traveling this way, it moves between an initial and a final position. Considering the motion of the object over this path, we want to identify which of these four graphs correctly shows its horizontal distance traveled over time. This means that for now, we’re not considering the fact that the object is also moving downward. We’re only taking into account its horizontal motion.
One important thing to realize about this horizontal motion is that once our object reaches its initial position, it doesn’t undergo any acceleration in this horizontal direction. That is, there is no force in the horizontal direction that tends to make the object speed up or slow down as it moves left to right. There is acceleration in the vertical direction due to gravity, but here we’re only considering horizontal motion. With no horizontal acceleration, we can expect the horizontal speed of our object to be constant as it moves from its initial to its final position.
Now let’s think about this. If the horizontal speed of our object is constant, that means that over equal intervals of time, it moves equal distances in this direction. For example, if our object moved, say, from here to here, from zero to one second of time, then from one to two seconds it would move that same distance, same thing with two to three seconds, and so on. If we call the distance that our object has moved over each of these one-second time intervals 𝑑, then we can see that at a time of zero seconds, our object hasn’t moved any distance horizontally at all. But then, by one second, it’s moved a distance 𝑑. At two seconds, it’s moved a distance of two 𝑑 and at three seconds, three 𝑑. Over time, the horizontal distance traveled is increasing in a linear pattern.
Looking now at our answer options, we see that in graph (a) the horizontal distance doesn’t change over time. If these were accurate, the initial position of our object and its final position would be along the same vertical line. We see they’re not, though. Over time our object does move horizontally. Its horizontal distance traveled changes. Both answer options (b) and (c) show horizontal distance starting off at a nonzero value when time is zero. But as we go from our initial to our final position, at first we have traveled a distance of zero horizontally. Our graph must begin at a horizontal distance of zero. This eliminates options (b) and (c).
And note that not only does option (d) begin at a horizontal distance of zero, that is, when time is zero, but it also demonstrates a linearly increasing distance traveled with time. This agrees with what we found earlier in studying this object’s horizontal motion. Therefore, for this part of our question, we choose answer option (d) as the graph that shows the horizontal distance moved by the object between its initial and final positions.
Let’s move on now to part two of our question.
Which of the graphs (e), (f), (g), and (h) shows the changes in the vertical distance moved by the object between its initial and final positions?
At this point, rather than considering horizontal motion like before, we’re thinking about the vertical motion of our object, specifically its vertical motion in moving between the initial and final positions. We noted that, unlike in the horizontal direction, in the vertical direction, our object does accelerate due to gravity. We would once again expect that at its initial position, the distance traveled by this object must be zero. Reviewing our answer options, we see that both options (e) and (g) show a vertical distance that is not zero at a time that is zero. This is not an accurate way of representing the vertical motion of this object, so we’ll cross out options (e) and (g).
Both of the graphs (f) and (h) show that the vertical distance traveled at a time of zero is zero. So now, we need to figure out whether vertical distance increases linearly with time, as in graph (h), or with an increasing slope over time, as in graph (f). To help us figure this out, we can recall that as our object moves between the initial and final positions, it’s undergoing what is called projectile motion. This motion is therefore described by what are sometimes called the kinematic equations of motion. One of these equations says that the distance an object travels as it undergoes projectile motion is equal to its initial velocity times the time elapsed plus one-half the acceleration the object experiences times the time elapsed squared.
Now let’s think about our object just as it leaves its initial position. At that instant, just as it begins to fall, its speed in the vertical direction is zero. Because our object’s initial speed in the vertical direction is zero, we can say that its distance travel is simply equal to one-half the acceleration it experiences times the time elapsed squared. We know what that acceleration is. It’s the acceleration due to gravity 𝑔. For our purposes, though, what’s important in this equation is this factor here, 𝑡 squared. This tells us that when it comes to vertical distance traveled and time passed, there’s not a linear relationship, but rather a quadratic relationship.
This means, for example, that if we were to double the time elapsed 𝑡, then the distance traveled 𝑑 would not double, but rather quadruple. This equation shows that there isn’t a linear relationship between vertical distance traveled and time passed. This means we won’t choose answer option (h). Rather, for our final answer, we’ll pick option (f). This shows that the object’s vertical distance traveled starts out at zero and then increases at an increasing rate as time passes. This reflects the fact that in the vertical direction, this object is accelerating.