### Video Transcript

A boy stands on a chair and throws
a ball vertically upward then catches it after it falls back downward. The boy’s friend stands on the
floor and watches. Which of the graphs, (a), (b), (c),
and (d), correctly shows the changes in kinetic energy, shown in red, and
gravitational potential energy, shown in blue, of the ball, measured from the
floor? The time axis of the graph starts
at the instant the ball leaves the boy’s hand. And the energy values cease to be
plotted at the instant the ball falls back to the height that it was released
from. Air resistance is negligible.

In this question, we’ve been asked
to select the graph that correctly shows how the energy of a ball changes when the
ball is thrown by someone standing on a chair. On each graph, the kinetic energy
of the ball is shown in red, and the gravitational potential energy of the ball is
shown in blue. Let’s start by thinking about what
we know about the energy of the ball and then see if we can narrow down our
options.

First, let’s think about the
gravitational potential energy of the ball. Recall that gravitational potential
energy is the category of energy associated with the height of an object above the
ground: the greater the height of the object, the greater its gravitational
potential energy. At the instant that the ball is
thrown, the ball is in the hand of a boy who is standing on a chair. So, before the ball is even thrown,
it will be at some height above the ground. This means that its initial
gravitational potential energy will be greater than zero.

When the ball is thrown vertically
upwards, its height increases and so does its gravitational potential energy. Eventually, the ball reaches its
maximum height, which is the point at which its gravitational potential energy is
also at a maximum. After this, the ball starts to fall
towards the ground. When the boy catches the ball, the
ball will have the same nonzero gravitational potential energy as it started
with.

If we look at the graphs we have
been given, we can see that the blue lines representing gravitational potential
energy all have similar shapes. The ball’s gravitational potential
energy increases after the ball is thrown until it reaches some maximum value and
then decreases until it reaches its initial value.

We can see that graphs (a), (b),
and (c) all show the ball starting with a gravitational potential energy that is
greater than zero. However, graph (d) shows the ball
starting with zero gravitational potential energy. This could only be the case if the
ball was thrown from ground level, which we know is not true. So, we can rule out graph (d).

Next, let’s think about the kinetic
energy of the ball. Recall that kinetic energy is the
energy category associated with the motion of an object; the faster an object moves,
the greater its kinetic energy. When the boy throws the ball, he
does work on it so that the ball has some initial kinetic energy. As the height of the ball
increases, its kinetic energy decreases.

When the ball reaches its maximum
height, there is an instant where it is completely stationary, just before it
changes direction and begins to fall back towards the ground. At the instant when the ball is
stationary, its kinetic energy is zero. As the ball falls towards the
ground, its kinetic energy increases. When the boy catches the ball, the
kinetic energy of the ball has the same value as it started with.

If we look at the graphs we have
left, we see that the red lines representing kinetic energy all have similar
shapes. The kinetic energy starts off at
some nonzero value, decreases until it reaches a minimum value, then increases
again, until it reaches the same value that it started with. In graphs (a) and (b), the minimum
value of the kinetic energy is equal to zero, just like we described before.

However, in graph (c), the kinetic
energy never reaches zero. Instead, it reaches some minimum
value that is greater than zero. We know this isn’t correct: the
ball must have zero kinetic energy when it reaches its maximum height and changes
direction. So, we know that graph (c) is not
correct, and we can rule this option out.

This leaves us with two graphs, (a)
and (b). To choose between these options, we
need to think about the energy transfers that take place while the ball is in the
air. So far, we have discussed the
changes in the ball’s gravitational potential and kinetic energy individually. However, the two quantities are
related. As the ball moves, its energy is
transferred back and forth between these two categories. When the ball’s height is
increasing, its energy is being transferred from kinetic energy to gravitational
potential energy. When the ball’s height is
decreasing, its energy is being transferred from gravitational potential energy to
kinetic energy.

We’re told that air resistance is
negligible. So, we’re safe to assume that these
are the only energy transfers that take place. This means that if the
gravitational potential energy of the ball increases, their kinetic energy must
decrease by the same amount. Similarly, if the kinetic energy
increases, the gravitational potential energy must decrease by the same amount. So, the change in both categories
of energy must be equal for the ball. Let’s think about what this means
in relation to the graphs.

Let’s start with graph (a). We can compare the initial energy
of the ball to its energy at this moment when it is halfway through its motion. Between these two times, the
gravitational potential energy of the ball has increased from this value to this
value. We can use an arrow to represent
the size of this change. Similarly, the kinetic energy of
the ball decreases from this value to this value. Again, we can represent the size of
this change with an arrow. These two arrows are the same
length. The increase in gravitational
potential energy is equal to the decrease in kinetic energy, just like we described
before.

If we repeat this process for graph
(b), we can see that the decrease in the ball’s kinetic energy is much greater than
the increase in its gravitational potential energy. In order for this to happen, some
of the ball’s kinetic energy would have to be transferred to some third energy
category. However, we know this isn’t the
case. The only energy transfers that
occur are between kinetic energy and gravitational potential energy. So, graph (b) cannot be the right
answer.

This leaves us with graph (a),
which correctly shows the changes in the ball’s kinetic and gravitational potential
energy. Graph (a) is therefore the correct
answer.