Question

Connect to the Big Idea The mechanical energy of a falling ball stays the same, even though it is constantly speeding up. Similarly, if you throw a ball upward, it slows down, even though its mechanical energy stays the same. Explain.

Short answer

Mechanical energy remains constant because potential and kinetic energy interchange while conserving the total energy.

Step-by-step solution

Understanding Mechanical Energy

In physics, mechanical energy is the sum of kinetic energy (energy of motion) and potential energy (energy due to position). For a falling object under gravity, the mechanical energy remains constant if we neglect air resistance. This means that the sum of kinetic and potential energy does not change.

Kinetic Energy Increase

As a ball falls, it loses height and thus its gravitational potential energy decreases. However, this energy is converted to kinetic energy because the speed of the ball increases while falling. The increase in kinetic energy exactly compensates for the decrease in potential energy, maintaining constant total mechanical energy.

Potential Energy Increase

Conversely, when you throw a ball upward, its kinetic energy (due to its speed) decreases as it climbs higher. However, as height increases, its potential energy increases. The decrease in kinetic energy is matched by an equal increase in potential energy, keeping the total mechanical energy constant.

Conservation of Mechanical Energy

The principle that explains the above transitions is the conservation of mechanical energy. It states that in the absence of non-conservative forces (like air resistance), the mechanical energy of a system remains constant. This conservation is why the energy stays the same for the ball throughout its motion, whether rising or falling.

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. When you hear someone talk about kinetic energy, they're referring to how fast something is moving and how that motion contributes to its energy. This type of energy is directly related to both the mass of the object and its velocity.
The formula for kinetic energy is given by: \[ KE = \frac{1}{2}mv^2 \] where \( m \) is the mass of the object and \( v \) is its velocity.
As a ball falls, it speeds up, meaning its velocity increases. As a result, the kinetic energy of the ball increases because of the change in velocity— the faster it falls, the more kinetic energy it gains.
It's fascinating to see that while the ball gains speed, something interesting is happening to its potential energy as well, leading to the concept of conservation of mechanical energy. Let's explore that by looking at potential energy next.

Potential Energy

Potential energy is the energy stored in an object due to its position or state. For a ball that is either falling or thrown upward, gravitational potential energy is the key focus. This energy depends on its height above the ground and the gravitational force acting upon it.
The formula for gravitational potential energy is: \[ PE = mgh \] where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height above the ground.
When a ball is at its highest point in the air, it has maximum potential energy because its height \( h \) is the greatest.

• As the ball falls, its height reduces, decreasing its potential energy.
• Conversely, when you throw a ball upwards, its height increases and so does its potential energy.

The fascinating part is how this decrease or increase in potential energy plays a role in changing kinetic energy, making it an integral part of understanding mechanical energy conservation.

Mechanical Energy

Mechanical energy is the sum of kinetic and potential energy in a system. This fundamental concept in physics explains why objects such as a falling or rising ball behave the way they do in a gravity field.
The principle known as the "conservation of mechanical energy" states that the total mechanical energy (kinetic + potential) of an object remains constant, provided no external forces like air resistance act on the object. This means: \[ ME = KE + PE = ext{constant} \]
When a ball falls, its potential energy decreases because its height decreases, but simultaneously its kinetic energy increases because it speeds up. Similarly, when a ball is thrown upwards, it slows down and gains height, decreasing its kinetic energy while increasing potential energy.

• This interchange of energy types keeps the total mechanical energy the same.
• Understanding this principle helps us grasp why objects behave predictably under the influence of gravity.

Without changing the total mechanical energy, energy transitions seamlessly between kinetic and potential forms, beautifully illustrating the conservation of energy in motion.