Problem 12 - Chemistry: The Central Science

Question

Suppose you toss a tennis ball upward. (a) Does the kinetic energy of the ball increase or decrease as it moves higher? (b) What happens to the potential energy of the ball as it moves higher? (c) If the same amount of energy were imparted to a ball the same size as a tennis ball, but of twice the mass, how high would it go in comparison to the tennis ball? Explain your answers.

Step-by-Step Solution

Answer

(a) The kinetic energy of the ball decreases as it moves higher, due to the deceleration caused by gravity acting against its motion. (b) The potential energy of the ball increases as it moves higher, as it gains height relative to the ground. (c) The tennis ball will go twice as high as the heavier ball because it reaches a higher potential energy at the same initial imparted energy.

Step 1: (a) Kinetic energy as the ball moves higher

When the ball is tossed upwards, it initially has a certain amount of kinetic energy due to its velocity. As it ascends, it will lose some velocity due to the force of gravity acting against its motion. Hence, its kinetic energy (given by \(KE = \frac{1}{2}mv^2\), where m is the mass and v is the velocity) will decrease as it moves higher.

Step 2: (b) Potential energy as the ball moves higher

As the ball ascends, it gains height relative to the ground. This increases its gravitational potential energy (given by \(PE = mgh\), where m is the mass, g is the acceleration due to gravity, and h is the height). Therefore, the potential energy of the ball increases as it moves higher.

Step 3: (c) Comparing the heights reached by the tennis ball and the heavier ball

We will use the law of conservation of energy to compare the heights reached by the tennis ball and the heavier ball. If both balls are initially given the same amount of energy, then at the highest point, all of that energy will be converted into potential energy. So we can set up an equation for each ball, equating the initial energy imparted (E) to the potential energy at the highest point: \(E = m_1gh_1\) for the tennis ball and \(E = m_2gh_2\) for the heavier ball. Since the problem states that the mass of the heavier ball is twice that of the tennis ball (\(m_2 = 2m_1\)), we can write the potential energy equations as follows: \(E = m_1gh_1\) and \(E = 2m_1gh_2\) Now, we can solve for the ratio of their heights (\(\frac{h_1}{h_2}\)): \[\frac{h_1}{h_2} = \frac{2m_1gh_2}{m_1gh_1}\] This simplifies to: \[\frac{h_1}{h_2} = \frac{2h_2}{h_1}\] Which means that the tennis ball will reach a height twice that of the heavier ball: \[h_1 = 2h_2\] So, the tennis ball will go twice as high as the heavier ball because it reaches a higher potential energy at the same initial imparted energy.

Key Concepts

Kinetic Energy

When you toss a tennis ball into the air, it starts with a certain amount of kinetic energy. Kinetic energy is the energy an object possesses because of its motion. For objects moving in a straight line like a thrown ball, kinetic energy is given by the formula:

• \(KE = \frac{1}{2}mv^2\)

Here, \(m\) refers to the mass of the ball, and \(v\) is its velocity.
As the ball moves upward against gravity, it slows down. Since velocity \(v\) decreases, the kinetic energy decreases as well. This is why the ball eventually stops at its peak height before descending back to the ground, where it gains velocity and hence, kinetic energy again. The decreasing kinetic energy signals that energy is being transferred to another form, namely potential energy.

Potential Energy

Potential energy is stored energy due to an object's position or condition. When the tennis ball rises, it gains height, which means it increases its potential energy. Specifically, the ball accumulates gravitational potential energy, described by:

• \(PE = mgh\)

In this equation, \(m\) stands for mass, \(g\) is acceleration due to gravity (approximately 9.8 m/s² on Earth), and \(h\) is the height above the ground.
As the ball ascends, \(h\) increases, thus increasing potential energy. This energy is ready to convert back into kinetic energy as the ball falls, highlighting the natural exchange between kinetic and potential energy governed by energy conservation principles.

Gravitational Force

The force that continuously acts on the tennis ball while it's in the air is gravitational force. Gravity pulls the ball downward, opposing its motion when it is tossed upwards. This force is crucial in understanding energy transformation during the ball's trajectory.
Gravitational force influences both kinetic and potential energy:

• Reduces kinetic energy as the ball ascends.
• Increases potential energy as the height increases when moving against gravity's direction.

Gravitational force ensures the interplay between motion (kinetic energy) and positional energy (potential energy), maintaining the system's total energy through its consistent pull.

Energy Conservation

Energy conservation, a fundamental concept in physics, states that energy cannot be created or destroyed. It can, however, change forms. In the tennis ball scenario, the concept of energy conservation explains how the ball’s energy shifts from kinetic to potential and back again during its flight.
Initially, all energy is kinetic when the ball is tossed upwards. As it rises and slows, kinetic energy diminishes while potential energy grows until it uses up all kinetic energy at the peak. On the way down, potential energy gets transformed back into kinetic energy.
Importantly, when two balls (one lighter and one heavier) are given the same energy, their maximum height differs. For the heavier ball, since the mass \(m\) is larger, its potential energy \(PE = mgh\) doesn't allow it to rise as high as the lighter tennis ball. Therefore, with equal energy supplies, the lighter ball climbs twice as high due to energy conservation across different masses.