An object is set in motion by an
initial force 𝐹 that acts diagonally upward, as shown in the diagram. The object undergoes projectile
motion. Which of the following graphs shows
the changes in the horizontal distance moved by the object between leaving the
ground and returning to the ground? (A), (B), (C), or (D).
In this question, we want to
determine the graph that correctly shows the changes in the horizontal distance of
the object during the projectile motion. First, we will look at the
diagram. The force acts diagonally
upwards. So this indicates that the object
has an initial horizontal velocity and an initial vertical velocity, which are both
nonzero. The question does not indicate the
presence of any force that acts horizontally on the object other than the force
responsible for the initial horizontal velocity of the object.
In addition, the acceleration of
the object throughout its motion is entirely vertical. So this means that the horizontal
component of the velocity will remain constant throughout the projectile motion. We can recall the equation speed
equals distance over time. So we know that distance is equal
to speed multiplied by time. Since the horizontal velocity is
constant, this means that the horizontal speed is constant. Since the horizontal speed remains
constant, then the equation tells us that the horizontal distance is directly
proportional to time.
Now that we know how the horizontal
distance changes with time, we can look at the graphs in the answer options.
Looking at graph A, we see that the
horizontal distance remains constant with time and always remains at the same
distance from the origin. This is not possible because this
suggests that the object did not move horizontally, meaning that the horizontal
speed is zero. We know that this is not true
because the object has a horizontal velocity provided by the initial force 𝐹. So graph (A) is incorrect.
In graph (B), we see that the
object starts at some horizontal distance away from the origin and then decreases
linearly with time. This is not correct because this
suggests that the object is approaching the point of launch. So graph (B) is incorrect.
In graph (C), we see that the
initial distance starts at zero, and the horizontal distance does increase linearly
with time. So graph (C) looks like the correct
Just to be sure, let’s look at
graph (D). In this graph, we see that the
object moves away a certain distance and then returns to the beginning. This is incorrect because the
object does not return to its point of origin. So graph (D) must also be
Therefore, we are left with graph
(C), which must be the correct answer. Graph (C) correctly shows the
changes in the horizontal distance moved by the object between leaving the ground
and returning to the ground.