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 these graphs shows the changes in the kinetic energy of 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 kinetic energy of the object during projectile motion.
First, we will look at the diagram. The force acts diagonally upwards. So this indicates that at point 𝐴, the object has an initial horizontal velocity and
an initial vertical velocity, which are both nonzero. Let’s consider the vertical velocity for now. The object undergoes projectile motion. So it accelerates downward uniformly throughout its entire motion due to the
gravitational acceleration 𝑔.
Uniform acceleration results in equal changes in speed in equal time intervals. So the initial vertical velocity of the object will decrease at a constant rate until
point 𝐵. Point 𝐵 is the highest point the object reaches. And at this point, the vertical velocity will be zero. After point 𝐵, the object will begin to fall back down to the ground due to
gravity. So the vertical velocity will decrease at the same constant rate as before.
Here, we note that we’re taking the upward direction as positive. And so the object falling back to the ground is in the downward direction, and the
vertical velocity is negative. So we know how the vertical velocity of the object changes during the projectile
motion. Now let’s consider the horizontal velocity.
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.
Now, we can recall that the equation for kinetic energy is given by half 𝑚𝑣
squared, where 𝑚 is the mass and 𝑣 is the velocity. The mass of the object does not change during the projectile motion. So the kinetic energy will be proportional to the velocity squared. Since the horizontal velocity remains constant during the projectile motion, there
will always be a nonzero amount of kinetic energy throughout the projectile
motion. This means we can immediately rule out graphs (A) and (D), since both show an initial
kinetic energy of zero.
If we now consider the component of kinetic energy coming from the vertical velocity,
the kinetic energy must decrease from point 𝐴 to point 𝐵, because the vertical
velocity also decreases from point 𝐴 to point 𝐵. The kinetic energy will decrease proportional to the vertical velocity squared. At point 𝐵, the kinetic energy will be at its minimum, because the vertical velocity
is equal to zero. But there is still some kinetic energy, because the horizontal velocity remains
constant at this point.
From point 𝐵 to point 𝐶, the vertical velocity becomes negative and decreases at a
constant rate. But since the kinetic energy is proportional to velocity squared, the kinetic energy
will be positive. So the kinetic energy will increase proportional to the vertical velocity
squared. We see that graph (B) corresponds to a kinetic energy that decreases linearly over
time, reaches a minimum value, and then increases linearly with time. We know that this graph is incorrect because the kinetic energy is proportional to
the velocity squared. So graph (B) is incorrect.
This leaves us with graph (C), which has a nonzero initial kinetic energy, decreases
quadratically, reaches a minimum value, and then increases quadratically. Therefore, graph (C) correctly shows the changes in the kinetic energy of the object
between leaving the ground and returning to the ground.