Video Transcript
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 gravitational potential 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 gravitational potential energy of the object during the projectile motion.
First, we will look at the diagram. The object is initially on the ground. So this means that the initial gravitational potential energy of the object is
zero. 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 object undergoes projectile motion. So it accelerates downward uniformly throughout its motion due to the gravitational
acceleration, 𝑔. This means that the acceleration of gravity, which is always downward, only acts on
the vertical velocity. And the horizontal velocity will always be constant throughout the time interval in
which the projectile is in the air, from the beginning.
At the beginning, the projectile will have kinetic energy due to the initial
velocity, and we know, by conservation of energy, that if there is no friction in
the air, the kinetic energy of the object will be transformed into gravitational
potential energy as the height increases. The kinetic energy of the object will initially decrease because the vertical
velocity decreases due to the gravitational acceleration 𝑔 acting downwards. This decrease in kinetic energy produces an increase in gravitational potential
energy. This will continue to happen until the object reaches the maximum height, where the
kinetic energy will be at its minimum and the gravitational potential energy will be
at its maximum.
Afterwards, the object will begin to fall back down to the ground. And so the gravitational potential energy will decrease and be transformed into
kinetic energy. Now that we know how the gravitational potential energy changes with time, we can
look at the graphs in the answer options.
Graph (A) represents a linear increase in gravitational potential energy to a maximum
value and then decreases linearly. However, we can recall that the kinetic energy is proportional to the velocity
squared. We also know that due to conservation of energy, if the kinetic energy of the object
decreases, the gravitational potential energy of the object must increase by the
same amount. So this means the gravitational potential energy should change quadratically with
time instead of linearly. So graph (A) is incorrect.
Graph (B) shows an initial nonzero gravitational potential energy and decreases
linearly until a minimum value and then increases linearly. This does not correctly describe the changes in gravitational potential energy of the
object. So graph (B) is incorrect.
Graph (C) also shows an initial nonzero gravitational potential energy and decreases
until a minimum value and then increases. This is incorrect, so graph (C) is incorrect.
This leaves us with graph (D), which shows an increase in potential energy that is
not linear over time to a maximum value and then decreases until it returns to the
ground. Therefore, the correct answer must be graph (D). Graph (D) correctly shows the changes in the gravitational potential energy of the
object between leaving the ground and returning to the ground.
