Question

An object moves with no friction or air resistance. Initially, its kinetic energy is \(10 \mathrm{~J}\), and its gravitational potential energy is \(30 \mathrm{~J}\). What is the greatest potential energy possible for this object? What is the greatest kinetic energy possible for this object?

Short answer

The greatest potential energy is 40 J, and the greatest kinetic energy is 40 J.

Step-by-step solution

Identify Total Mechanical Energy

The total mechanical energy of a system is conserved when there are no external forces like friction or air resistance. The total mechanical energy (E) is the sum of kinetic energy (KE) and gravitational potential energy (PE). Initially, the object's KE is 10 J and its PE is 30 J, so the total energy is 10 J + 30 J = 40 J.

Calculate the Greatest Potential Energy

Since the total energy is conserved and equals 40 J, the greatest potential energy occurs when the kinetic energy is zero. At this point, the potential energy must equal the total energy. Thus, the greatest potential energy is 40 J.

Calculate the Greatest Kinetic Energy

Similarly, the greatest kinetic energy occurs when the potential energy is zero. Again, since the total energy is conserved, the greatest kinetic energy also equals the total energy, which is 40 J.

Key concepts

Understanding Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It depends on two main factors: the mass of the object and its velocity. The formula to calculate kinetic energy (\( KE \) ) is:
\( KE = \frac{1}{2} m v^2 \)

• **Mass (\( m \) )**: This is how much matter is contained in the object, measured in kilograms.
• **Velocity (\( v \) )**: The speed of the object in a specific direction, measured in meters per second.

As you can see from the formula, even a small increase in velocity leads to a significant increase in kinetic energy because the velocity term is squared.
In cases where there’s no friction or air resistance, like in the given problem, the kinetic energy can vary, but the total mechanical energy will remain constant. This means when the velocity is at its highest, the kinetic energy will be at its greatest, provided no potential energy is utilized.

Exploring Potential Energy

Potential energy is the energy stored in an object due to its position in a force field, commonly a gravitational field. For an object held at a height above the ground, the gravitational potential energy (\( PE \) ) is given by:
\( PE = mgh \)

• **Mass (\( m \) )**: Like kinetic energy, mass here is in kilograms.
• **Gravitational acceleration (\( g \) )**: A constant on Earth, approximately \( 9.81 \text{ m/s}^2 \) .
• **Height (\( h \) )**: The height of the object above a reference point, measured in meters.

This formula shows that an object gains potential energy as it is lifted higher.
In situations where no energy is lost to external forces, potential energy can transform into kinetic energy and vice versa, depending on the movement and position of the object. In the given problem, the greatest potential energy is reached when the object stops moving (kinetic energy is zero), which would occur at a maximum height.

Mechanical Energy Conservation

The principle of mechanical energy conservation states that the total mechanical energy in a closed system remains constant, provided no non-conservative forces (like friction or air resistance) act on the system.
This total mechanical energy (\( E \) ) is the sum of kinetic energy (\( KE \) ) and potential energy (\( PE \) ):
\( E = KE + PE \)
In a frictionless environment, if an object's kinetic energy decreases, its potential energy increases by an equal amount, and vice versa. This concept is what allows us to predict the highest potential and kinetic energies. In the original exercise, both types of energy can reach 40 J because the total is conserved:

• **Greatest Kinetic Energy**: Achieved when all energy is kinetic, potential energy is zero.
• **Greatest Potential Energy**: Achieved when all energy is potential, kinetic energy is zero.

Understanding this conservation principle helps in analyzing how energy transfers and transforms in various physics problems, ensuring no energy is lost, only transformed.