Question

A baseball is dropped from the top of a building. Air resistance acts on the baseball as it drops. Which of the following statements is true? a) The change in potential energy of the baseball as it falls is equal to the kinetic energy of the baseball just before it strikes the ground. b) The change in potential energy of the baseball as it falls is greater than the kinetic energy of the baseball just before it strikes the ground. c) The change in potential energy of the baseball as it falls is less than the kinetic energy of the baseball just before it strikes the ground. d) The change in potential energy of the baseball is equal to the energy lost due to the friction from the air resistance while the ball is falling.

Short answer

a) The change in potential energy of the baseball as it falls is equal to the kinetic energy of the baseball just before it strikes the ground. b) The change in potential energy of the baseball as it falls is greater than the kinetic energy of the baseball just before it strikes the ground. c) The change in potential energy of the baseball as it falls is less than the kinetic energy of the baseball just before it strikes the ground. Answer: b) The change in potential energy of the baseball as it falls is greater than the kinetic energy of the baseball just before it strikes the ground.

Step-by-step solution

Analyze the energy changes when a baseball falls without air resistance

When a baseball falls without air resistance, it is subjected to only gravitational force acting on it. In this case, the potential energy of the baseball is converted into kinetic energy. According to the conservation of mechanical energy principle, the change in potential energy should be equal to the kinetic energy gained by the baseball before it strikes the ground.

Analyze the energy changes when a baseball falls with air resistance

When a baseball falls with air resistance, it loses some of its energy due to friction from the air resistance. In this case, not all the potential energy is converted into kinetic energy. Some of it is used to overcome the air resistance.

Compare the potential energy change and kinetic energy of the baseball

As some of the potential energy is used to overcome air resistance, the change in potential energy should be greater than the kinetic energy of the baseball just before it strikes the ground.

Choose the correct statement

Based on our analysis in step 3, the correct statement among the given options is: b) The change in potential energy of the baseball as it falls is greater than the kinetic energy of the baseball just before it strikes the ground.

Key concepts

Conservation of Mechanical Energy

In physics, the conservation of mechanical energy is an important principle that states in a closed system with no external forces acting, the total mechanical energy remains constant. This means that energy is neither created nor destroyed, but it can be transformed from one form to another. Mechanical energy is usually considered as the sum of potential energy and kinetic energy.

When an object moves in a gravity field without any non-conservative forces like air resistance, its mechanical energy is conserved. For example, a pendulum at its highest point has maximum potential energy and as it swings down, this potential energy is converted into kinetic energy. If there is no air resistance, the speed of the pendulum will be greatest at the lowest point, where potential energy is at its lowest.

Potential Energy

The term potential energy refers to the energy that is stored within an object due to its position or configuration. Gravitational potential energy is the most common type encountered in high school and introductory college physics courses. This type of potential energy is dependent on the height of an object above a reference level.

Mathematically, gravitational potential energy, often denoted as PE or U, can be expressed with the formula: \( PE = m \cdot g \cdot h \), where 'm' is mass, 'g' is the acceleration due to gravity, and 'h' is the height of the object. When an object is raised, its potential energy increases, and when it falls, the potential energy decreases, typically converting into kinetic energy if no other forces are present.

Kinetic Energy

On the other hand, kinetic energy is the energy of motion. An object possesses kinetic energy if it is moving. The faster an object moves, the more kinetic energy it has. The equation for kinetic energy (KE) is: \( KE = \frac{1}{2} m v^2 \), where 'm' is mass and 'v' is the velocity of the object.

Imagine sliding down a hill on a sled: at the top, you have potential energy, and as you slide down, that potential energy is converted into kinetic energy, causing you to pick up speed. In an ideal scenario without friction, your kinetic energy at the bottom would be equal to the potential energy you had at the top.

Air Resistance

In real-world scenarios, air resistance, also known as drag, plays a significant role in the motion of objects. It is a force caused by air, which opposes the motion of an object through it. As objects move through the air, they collide with air molecules, which creates a force that slows down their movement.

The effect of air resistance is notably observed on objects that have a large surface area or those at high speeds, like parachutes or meteorites entering the atmosphere. Even a baseball falling off a building experiences air resistance, which reduces its kinetic energy below what would be expected if the motion were only subject to gravitational forces. Therefore, when considering the conservation of mechanical energy, the work done against air resistance must be accounted for, leading to a decrease in the object's mechanical energy.