Video Transcript
Which of the graphs (a), (b), (c),
and (d) correctly shows the changes in kinetic energy, shown in red, and the
gravitational potential energy, shown in blue, for a ball being thrown vertically
upward and falling back to Earth? The time axis of the graph starts
at the instant the ball leaves the thrower’s hand. And the energy values cease to be
plotted at the instant that the ball falls back to the height that it was released
from. Air resistance is negligible.
In this question, we are asked to
identify the graph that correctly shows how the energy of a ball changes when it is
thrown. Before we consider the graphs that
we’ve been given, let’s see if we can work out anything about what the correct graph
should look like. We’ll start by thinking about the
ball’s gravitational potential energy.
Recall that gravitational potential
energy is the category of energy associated with the height of an object above the
ground. The higher an object is, the
greater its gravitational potential energy.
When a ball is thrown, it starts at
its lowest point. And hence, its gravitational
potential energy is at a minimum. We can represent this on an
energy-versus-time graph of our own, by drawing a point here. After the ball is thrown 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. On our graph, this point is
here.
After this, the ball begins to fall
back down towards the ground. As its height decreases, so does
its gravitational potential energy. When the ball returns to the height
that it was thrown from, its gravitational potential energy returns to its initial
value. We can add this to our graph by
drawing a point here.
Next, let’s think about the kinetic
energy of the ball. When the ball is thrown, the
thrower does work on the ball. This means that, initially, the
kinetic energy of the ball is greater than zero. We can draw this on our graph with
purple, here. As the ball’s height increases, its
kinetic energy is transferred to gravitational potential energy. This means that the kinetic energy
of the ball decreases. When the ball reaches its maximum
height, it is momentarily stationary and the kinetic energy of the ball is zero. We can add this to our graph by
drawing a point here.
This corresponds to the same time
at which the ball has the maximum gravitational potential energy. As the ball begins to fall towards
the ground, gravitational potential energy is transferred to kinetic energy, causing
the kinetic energy of the ball to increase. When the ball reaches the height it
was thrown from, its kinetic energy returns to its initial value. So, we can add one final point to
our graph, here.
Now, we can compare the points we
have just plotted to graphs given to us by the question. We can see that this pattern of
points is consistent with all of the graphs, except for graph (b). Graph (b) shows both categories of
the ball’s energy crossing below the horizontal axis and becoming negative. This is not possible. A quantity of energy cannot have a
value that is below zero. So, we can rule out graph (b).
Now let’s think about graphs (a),
(c), and (d). All of the graphs are consistent
with the points that we plotted earlier, but we can see that the curves on the
graphs have different shapes. In graph (a), both the kinetic and
gravitational potential energy are represented by a smooth curve. In graph (c), the gravitational
potential energy has a smooth curve, but the kinetic energy has a sharp point at its
minimum. In graph (d), the kinetic energy
has a smooth curve, but the gravitational potential energy has a sharp point at its
maximum.
So, how do we decide which graph
shows the curves with the correct shape? The key is to remember that during
the ball’s motion, its energy is transferred between the categories of kinetic
energy and gravitational potential 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. When the kinetic energy decreases,
the gravitational potential energy increases by the same amount. Similarly, when the gravitational
potential energy decreases, the kinetic energy increases by the same amount. Let’s see if we can apply this
concept to these graphs.
We can start with graph (a). Let’s look at the energy changes
shown on graph (a) between these two times. We can see that the gravitational
potential energy increases during this time, from this value to this value. The size of this increase can be
represented by the length of this arrow. The kinetic energy of the ball
decreases during this time, from this value to this value. Again, we can represent this
decrease using an arrow.
We can see that these two arrows
are the same length. This means that the increase in the
gravitational potential energy is equal to the decrease in the kinetic energy. This is just what we would expect
when energy is being transferred from kinetic energy to gravitational potential
energy, like we described before.
But if we repeat this process for
graph (c), we can see that the decrease in kinetic energy is much greater than the
increase in gravitational potential energy. We know that energy is always
conserved. So, this cannot be the correct
answer. Similarly for graph (d), the
increase in gravitational potential energy is much greater than the decrease in
kinetic energy. So, we can rule out this option,
too.
This leaves us with graph (a),
which correctly shows the changes in the ball’s kinetic and gravitational potential
energy. Graph (a) is the correct
answer.
