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.

Short answer

(a) The kinetic energy of the ball decreases as it moves higher because its velocity decreases due to gravity. (b) The potential energy of the ball increases as it moves higher as the height 'h' increases in the potential energy formula. (c) The tennis ball will go twice as high as the ball with twice the mass when the same amount of energy is imparted to them.

Step-by-step solution

Part (a): Kinetic energy change

As a tennis ball is tossed upward, its velocity decreases due to the force of gravity acting on it. Kinetic energy is calculated as \(KE = \frac{1}{2}mv^2\), where 'm' is the mass of the ball, and 'v' is its velocity. Since the velocity of the ball is decreasing as it moves higher, the kinetic energy of the ball will also decrease.

Part (b): Potential energy change

As the tennis ball rises higher, its gravitational potential energy increases. Potential energy is calculated as \(PE = mgh\), where 'm' is the mass, 'g' is the acceleration due to gravity, and 'h' is the height. As the ball moves higher, 'h' increases, meaning the potential energy of the ball increases.

Part (c): Comparing maximum heights

When the same amount of energy is imparted to both balls, the initial total energy of each ball is equal. As the balls rise, they lose kinetic energy and gain potential energy. When they reach their maximum heights, their kinetic energy becomes zero, and the entire initial energy has been converted into gravitational potential energy. Let the initial total energy of the balls be E, and the maximum heights reached by the tennis ball and the heavier ball be \(h_t\) and \(h_2\) respectively. For the tennis ball, we have, \(E = m_1gh_t\). For the twice-mass ball, we have, \(E = 2m_1gh_2\), where \(m_1\) is the mass of the tennis ball. We are interested in finding the ratio of the heights, i.e., \(\frac{h_t}{h_2}\). We can divide the potential energy equations thus: \[\frac{m_1gh_t}{2m_1gh_2} = \frac{h_t}{h_2}\] Which simplifies to: \[\frac{h_t}{h_2} = \frac{2}{1}\] So the tennis ball will go twice as high as the ball with twice the mass, given that the same amount of energy is imparted to them.

Key concepts

Energy Conservation in Physics

The principle of energy conservation is a cornerstone in physics. It states that energy cannot be created or destroyed in an isolated system; it can only be transformed from one form to another. This principle is applied every day in various fields, from engineering to environmental science, and it has concrete implications for our tennis ball exercise.

In the exercise, as the tennis ball is tossed upwards, the energy imparted to the ball by the throw is conserved. Initially, most of this energy is in the form of kinetic energy—the energy it possesses due to its motion. As the ball rises and slows down due to gravity, energy conservation dictates that the kinetic energy must be converted into another form, which in this case, is gravitational potential energy. At the peak of the ball's trajectory, the kinetic energy is zero, and all the energy exists as potential energy. When the ball starts descending, the process reverses, and the potential energy is transformed back into kinetic energy. This phenomenon is a perfect example of the conservation of energy in action.

Gravitational Potential Energy

Gravitational potential energy (PE) is the energy that an object possesses due to its position in a gravitational field. The higher an object is above the ground, the more potential energy it has. The key variables in the calculation of gravitational potential energy are the object's mass (m), the height above ground (h), and the acceleration due to gravity (g), which on Earth is approximately 9.81 m/s^2.

The formula for gravitational potential energy is given by \(PE = mgh\). For the tennis ball exercise, as the ball moves higher, its height (h) increases. This increase in height means the ball's potential energy increases as well. It's worth noting that the ball's mass or the gravitational constant doesn't change; only the height does. So, as we saw in the exercise, when the ball is at its maximum height, the kinetic energy is completely converted to potential energy, fully illustrating the transfer of energy within the system due to gravity.

Kinetic Energy Formula

Kinetic energy is directly associated with motion. An object in motion, whether it's a speeding car or a rolling ball, has kinetic energy. The formula to calculate kinetic energy (KE) is \(KE = \frac{1}{2}mv^2\), where 'm' stands for mass and 'v' stands for the velocity of the object.

In our exercise, the tennis ball begins with a certain amount of kinetic energy once it's tossed. As the ball rises, gravity slows it down, reducing its velocity and, consequently, its kinetic energy. This deceleration continues until the ball reaches its apex, where the velocity is zero, and accordingly, its kinetic energy is also zero. Here, we can also touch upon the improved understanding of mass's role in kinetic energy. A ball with twice the mass, given the same amount of kinetic energy, would reach a lower height because the kinetic energy would also need to account for the increased mass, according to the formula. Thus, despite the same kinetic push, the ball with greater mass wouldn't climb as high.