Energy Transformations

This section explores the transformations of energy between kinetic, elastic potential, and gravitational potential energy, as well as energy dissipated to the environment.

1. Kinetic Energy

• Definition: The energy an object possesses due to its motion.
• Formula (at low speeds): $E_k = \frac{1}{2}mv^2$
  • Where:
    • $E_k$ is kinetic energy (Joules, J)
    • $m$ is mass (kilograms, kg)
    • $v$ is velocity (meters per second, m/s)

KEY TAKEAWAY: Kinetic energy is directly proportional to mass and the square of velocity. A small change in velocity results in a large change in kinetic energy.

2. Elastic Potential Energy

• Definition: The energy stored in a deformable object (like a spring) when it is stretched or compressed.
• Hooke’s Law: $F = -kx$
  • Where:
    • $F$ is the restoring force (Newtons, N)
    • $k$ is the spring constant (Newtons per meter, N/m) - a measure of the spring’s stiffness
    • $x$ is the displacement from the equilibrium position (meters, m)
      • The negative sign indicates that the restoring force opposes the displacement.
• Elastic Potential Energy Formula: $E_s = \frac{1}{2}kx^2$
  • Derivation: Elastic potential energy is the area under the force-distance graph for a spring obeying Hooke’s Law. Since $F = kx$, the graph is a straight line, and the area under the line (a triangle) is $\frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2} \cdot x \cdot kx = \frac{1}{2}kx^2$.
• Force-Distance Graph: The area under the force-distance graph represents the work done to stretch or compress the spring, which is equal to the elastic potential energy stored in the spring.

EXAM TIP: Be careful with units. Ensure displacement is in meters before calculating elastic potential energy.

3. Gravitational Potential Energy

• Definition: The energy an object possesses due to its position in a gravitational field.
• Formula (near Earth’s surface): $E_g = mg\Delta h$
  • Where:
    • $E_g$ is gravitational potential energy (Joules, J)
    • $m$ is mass (kilograms, kg)
    • $g$ is the acceleration due to gravity (approximately 9.8 m/s² on Earth)
    • $\Delta h$ is the change in height (meters, m) relative to a reference point.
• Force-Distance Graph: The area under a force-distance graph also represents gravitational potential energy. If the force is constant (like near the Earth’s surface), the area is simply $F \cdot \Delta h = mg \Delta h$.
• Field-Distance Graph: The area under a field-distance graph (gravitational field strength vs. distance) multiplied by the mass of the object gives the change in gravitational potential energy.

COMMON MISTAKE: Gravitational potential energy is relative to a chosen zero point. The change in gravitational potential energy is what matters in most problems.

4. Energy Dissipation

• Definition: The process by which energy is converted into forms that are not easily recoverable or used to do work. This energy is often dissipated as heat, sound, or through deformation of materials.
• Forms of Dissipated Energy:
  • Heat: Friction between surfaces, air resistance, and internal friction within materials all generate heat.
  • Sound: Collisions and vibrations can produce sound waves, which carry energy away from the system.
  • Deformation: Permanent changes in the shape of an object during a collision or interaction consume energy.
• Impact on Conservation of Energy: While the total energy of a closed system is always conserved, energy dissipation means that the amount of energy in the forms we are interested in (kinetic, potential) decreases over time.

STUDY HINT: Think about real-world scenarios. Where does the “lost” energy go when a bouncing ball eventually stops?

5. Elastic and Inelastic Collisions

• Collision: An event in which two or more objects exert forces on each other for a relatively short period of time.
• Elastic Collision: A collision in which kinetic energy is conserved.
  • In a perfectly elastic collision, no energy is dissipated as heat, sound, or deformation.
  • Momentum is always conserved in collisions (elastic or inelastic) in a closed system.
• Inelastic Collision: A collision in which kinetic energy is not conserved.
  • Some kinetic energy is converted into other forms of energy, such as heat, sound, or deformation.
  • Momentum is still conserved.
• Perfectly Inelastic Collision: A special case of inelastic collision where the objects stick together after the collision.

Comparing Elastic and Inelastic Collisions

REMEMBER: “Elastic” means kinetic energy is conserved. Think “elastic band” - it bounces back with no loss of energy (ideally).

6. Conservation of Energy

• Principle: In a closed system, the total amount of energy remains constant. Energy can be transformed from one form to another, but it cannot be created or destroyed.
• Equation: $E_{\text{initial}} = E_{\text{final}}$
  • This means the sum of all forms of energy (kinetic, potential, thermal, etc.) at the beginning of a process is equal to the sum of all forms of energy at the end.
• Applying Conservation of Energy:
  1. Identify the system and its initial and final states.
  2. List all forms of energy present in the initial and final states.
  3. Account for any energy dissipation (e.g., work done by friction).
  4. Set up an equation equating the initial and final energies and solve for the unknown.

APPLICATION: Roller coasters use conservation of energy to convert between gravitational potential energy and kinetic energy.

Examples

1. A ball dropped from a height: Gravitational potential energy is converted to kinetic energy as the ball falls. Some energy is dissipated as heat and sound upon impact with the ground. If the ball bounces back up, some of the kinetic energy is converted back into gravitational potential energy, but the bounce will be lower due to energy losses.
2. A spring-mass system: When a mass attached to a spring is displaced and released, energy is continuously transformed between elastic potential energy (when the spring is stretched or compressed) and kinetic energy (when the mass is moving). Due to damping forces (like air resistance), some energy will be dissipated as heat.
3. A car crash: Kinetic energy of the moving vehicles is converted into various forms of energy during the collision, including heat (due to friction and deformation), sound (the crash noise), and deformation (damage to the vehicles). This is an inelastic collision.

VCAA FOCUS: VCAA often tests your ability to apply conservation of energy to solve problems involving multiple energy transformations. Pay close attention to identifying all forms of energy involved and accounting for energy dissipation.