Question

An object moves with no friction or air resistance. Initially, its kinetic energy is \(10 \mathrm{~J}\), and its gravitational potential energy is \(20 \mathrm{~J}\). What is its kinetic energy when its potential energy has decreased to \(15 \mathrm{~J}\) ? What is its potential energy when its kinetic energy has decreased to 5 J?

Short answer

Kinetic energy is 15 J and potential energy is 25 J.

Step-by-step solution

Understanding Energy Conservation

The total mechanical energy of the object is conserved since there is no friction or air resistance. This means that the sum of kinetic energy (KE) and potential energy (PE) remains constant. Initially, the total energy is a sum of the initial kinetic energy and potential energy, which is \(10 \mathrm{~J} + 20 \mathrm{~J} = 30 \mathrm{~J}\).

Calculating Kinetic Energy when PE Decreases

When the gravitational potential energy decreases to \(15 \mathrm{~J}\), the total energy is still \(30 \mathrm{~J}\). Since total energy is conserved, the kinetic energy is the difference between the total energy and the new potential energy: \(30 \mathrm{~J} - 15 \mathrm{~J} = 15 \mathrm{~J}\).

Calculating Potential Energy when KE Decreases

When the kinetic energy decreases to \(5 \mathrm{~J}\), the total energy remains \(30 \mathrm{~J}\). The potential energy is the difference between the total energy and the new kinetic energy: \(30 \mathrm{~J} - 5 \mathrm{~J} = 25 \mathrm{~J}\).

Key concepts

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. It depends on the mass of the object and the velocity at which it moves. The formula for kinetic energy is given by \[ KE = \frac{1}{2} m v^2 \]where:

• \( KE \) is the kinetic energy,
• \( m \) is the mass of the object,
• \( v \) is the velocity of the object.

Kinetic energy varies directly with the square of the velocity, meaning that even small increases in velocity can lead to significant increases in kinetic energy.
This concept can be easily seen in our exercise. Initially, the object had a kinetic energy of \(10 \mathrm{~J}\), and as the potential energy dropped, kinetic energy increased to \(15 \mathrm{~J}\). With no external forces acting, energy is just converting from one form to another.

Potential Energy

Potential energy is the stored energy of an object due to its position or state. For objects influenced by gravity, this energy is referred to as gravitational potential energy. The formula to calculate gravitational potential energy is:\[ PE = mgh \]where:

• \( PE \) is the potential energy,
• \( m \) is the mass of the object,
• \( g \) is the acceleration due to gravity,
• \( h \) is the height above a reference point.

This type of energy is crucial for systems experiencing height changes, like in our problem where the potential energy started at \(20 \mathrm{~J}\), then decreased as the object moved to a lower gravitational position, eventually turning into kinetic energy.
Later, when kinetic energy decreased to \(5 \mathrm{~J}\), the potential energy rose to \(25 \mathrm{~J}\), showcasing how these energies swap roles but remain part of the total mechanical energy.

Mechanical Energy

Mechanical energy is the sum of kinetic and potential energy in a system. It measures the total energy from motion and position. Unlike thermal energy or chemical energy, mechanical energy is visible in everyday actions, like throwing a ball or lifting a book.
In a closed, isolated system with no external forces like friction, mechanical energy is conserved, meaning it remains constant throughout the interaction. This is apparent in our exercise, where the total mechanical energy continued to be \(30 \mathrm{~J}\) regardless of the individual fluctuations between kinetic and potential energy.
Understanding mechanical energy conservation helps in predicting an object's behavior. For instance, if the potential energy reduces, we can instantly determine the increase in kinetic energy, ensuring the total remains constant.