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 vertical speed of the object between its initial and final positions?
Looking at our diagram, we see our object with this force 𝐅 acting on it. After experiencing this force, the object moves across a smooth horizontal surface and then begins projectile motion. Considering its movement between this initial and this final position, we want to identify which of these four graphs accurately models the vertical speed of the object over time. That is, we’re to consider this component of the object’s motion. The first thing we can say is, at the outset at its initial position, the vertical speed of this object is zero. That’s because it hasn’t quite yet started moving due to its fall. Right away then, we can cross out any answer choices that don’t have a vertical speed of zero at a time of zero. We see that options (a) and (c) both include nonzero initial vertical speeds.
The question then becomes, does this object as it falls have a steadily increasing vertical speed? Or does its vertical speed increase at an increasing rate? To begin figuring this out, let’s recall that as our object moves from its initial to its final position, it undergoes projectile motion. This means that a set of equations, sometimes called the kinematic equations of motion, apply to describing the object’s movement. One of these equations of motion says that the final velocity of an object undergoing projectile motion is equal to its initial velocity plus its acceleration times the time elapsed. This equation does involve a velocity rather than speed, that is, vectors rather than a scalar.
But we can recall the connection between velocity and speed that the magnitude of velocity equals speed. This equation then can help us understand how the vertical speed of our object changes over time. Notice that this equation is a linear equation. That is, every factor in it is effectively raised to the first power. This means, for example, that if 𝑣 zero were zero, in which case the equation would look like this, then we could change the right-hand side of this equation, say, by doubling the time 𝑡. And that would make our final velocity change by the same factor. This suggests, and it’s indeed the case, that there is a linear relationship between the final velocity of our object and the time passed 𝑡. If 𝑡 is doubled, so is velocity; if 𝑡 is tripled, so is velocity.
Therefore, when it comes to speed against time, in particular, vertical speed against time, we know that a change by some factor in one of these variables will lead to a change by the same factor in the other. This indicates that graph (b), which does not show a linear relationship between vertical speed and time, can’t be correct. Because the vertical acceleration of our object in projectile motion is constant, the rate at which its vertical speed changes over time is constant too. Our answer is graph (d).
Let’s look now at part two of this question.
Which of the graphs (e), (f), (g), and (h) shows the changes in the horizontal speed of the object between its initial and final positions?
Okay, so now, rather than considering the vertical motion of our object, we’re thinking about its horizontal motion. Once again, we can use this equation to help us understand our object’s speed. Notice, though, that there’s a difference between using it for vertical and horizontal motion. In the vertical direction, we have a nonzero acceleration. Horizontally, though, there is no acceleration for our object. This means that we can substitute zero in for 𝑎. And that makes the equation simplify to 𝑣 sub f equals 𝑣 sub zero. This equation is true for the horizontal motion of a projectile. It says that the velocity of that projectile is always the same. This tells us that its speed is constant as well.
So then, we’re looking for a graph that shows us a constant horizontal speed over time. A constant speed is one that doesn’t change. And here in graph (e), we see just this thing. Notice also that this horizontal speed initially is not zero. We would expect that because our object was given a push and moved across a smooth horizontal surface before beginning projectile motion. This means it is already moving left to right at some nonzero speed. We don’t know exactly what that speed is. But for our purposes, the important thing is we know that it stays the same over the object’s motion.
The graph that shows the change in horizontal speed of the object between its initial and final positions is graph (e).