Question Video: Identifying Changes in Energy for a Ball Thrown Vertically Upwards | Nagwa Question Video: Identifying Changes in Energy for a Ball Thrown Vertically Upwards | Nagwa

Question Video: Identifying Changes in Energy for a Ball Thrown Vertically Upwards Physics • First Year of Secondary School

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.

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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.

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