Question

The potential energy of an object decreases by \(10 \mathrm{~J}\). What is the change in the object's kinetic energy, assuming there is no friction in the system?

Short answer

The object's kinetic energy increases by 10 J.

Step-by-step solution

Understand the Conservation of Energy Principle

The principle of conservation of mechanical energy states that in an isolated system with no non-conservative forces like friction, the total mechanical energy (sum of potential and kinetic energy) of the system remains constant. This implies that any change in potential energy should result in an equal and opposite change in kinetic energy.

Determine the Change in Potential Energy

According to the problem, the potential energy of the object decreases by 10 J. This change can be expressed as \( \Delta PE = -10 \mathrm{~J} \).

Apply the Conservation of Energy

Since no external non-conservative forces are acting on the system, the change in kinetic energy \( \Delta KE \) will be equal in magnitude but opposite in sign to the change in potential energy: \[ \Delta KE = - \Delta PE \]

Calculate the Change in Kinetic Energy

Substitute the given change in potential energy into the equation: \[ \Delta KE = -(-10 \mathrm{~J}) = 10 \mathrm{~J} \] Thus, the object's kinetic energy increases by 10 J.

Key concepts

Potential Energy

Potential energy is the stored energy in an object due to its position or state. Imagine holding a ball at a height above the ground. The ball has potential energy because it can fall.

• Potential energy is often denoted by PE.
• It depends on factors like height, mass, and gravity, especially in the context of gravitational potential energy.
• 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.

When you let go of the ball, it falls, and its potential energy decreases. In our exercise, the potential energy decrease of 10 J means this energy will transform to another form—kinetic energy, ensuring energy is conserved.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion. Whenever something moves, it has kinetic energy, which is often abbreviated as KE.

• The faster an object moves, the more kinetic energy it has.
• It also depends on the object's mass.
• The formula to calculate kinetic energy is \( KE = \frac{1}{2}mv^2 \), where \( m \) is mass and \( v \) is velocity.

In the discussed problem, when the potential energy drops by 10 J, kinetic energy increases by the same amount, 10 J, if no energy is lost to friction. This conversion keeps the total energy of the system unchanged.

Mechanical Energy

Mechanical energy is the sum of potential and kinetic energy in a system. It explains the total energy an object possesses due to its state and movement.

• Mechanical energy is conserved in isolated systems without external influences like friction or air resistance.
• The conservation principle states \( ME_{initial} = ME_{final} \), meaning the sum of PE and KE remains constant.
• This principle is crucial for understanding energy transformation during processes like the swinging of a pendulum or falling objects.

In the exercise scenario, the decrease in potential energy fully converts into kinetic energy. Thus, mechanical energy remains constant at all times, assuring no energy loss from the system.