Conservation of Energy: Comprehensive Notes

Lesson 3: Conservation of Energy

Focus Question

  • Can energy be lost?

New Vocabulary

  • Law of conservation of energy: Energy cannot be created or destroyed in a closed, isolated system.
  • Mechanical energy (ME): The sum of kinetic and potential energy in a system.
  • Elastic collision: A collision where kinetic energy is conserved.
  • Inelastic collision: A collision where kinetic energy decreases.

Review Vocabulary

  • Closed system: A system that does not exchange mass with its surroundings.

The Law of Conservation of Energy

  • The law of conservation of energy states that in a closed, isolated system, energy can neither be created nor destroyed; rather, energy is conserved.
  • The sum of the kinetic energy and the potential energy of the objects in a system is the system’s mechanical energy (ME).
  • ME = KE + PE
  • Kinetic energy includes both translational and rotational kinetic energies.
  • Potential energy includes the gravitational and elastic potential energies.
  • In some situations, mechanical energy is conserved. In others, some of the mechanical energy is transformed into other forms, such as thermal energy/heat.

Conservation of Mechanical Energy

  • KE1 + PE1 = KEf + PEf
    • Where:
      • KE_1 = Initial Kinetic Energy
      • PE_1 = Initial Potential Energy
      • KE_f = Final Kinetic Energy
      • PE_f = Final Potential Energy

Energy Bar Diagrams

  • Visual representation of energy distribution in a system.

Examples of Conservation of Energy

Roller Coasters

  • At the top of the hill (A), the roller coaster has maximum gravitational potential energy (GPE) and minimal kinetic energy (KE).
  • As it descends (B, C), GPE converts to KE, increasing speed.
  • At the bottom (D), KE is maximum, GPE is minimum.
  • As it ascends again (E), KE converts back to GPE.

Skiing

  • At the top of the slope (A), the skier has maximum GPE.
  • As the skier goes down the slope (B, C), GPE is converted to KE, increasing speed.

Pendulums

  • At the highest points (A, E), the pendulum has maximum GPE and minimum KE.
  • At the lowest point (C), the pendulum has maximum KE and minimum GPE.
  • During the swing (B, D), there is a continuous conversion between GPE and KE.

Example Problem: Diver

  • A 68.2-kg diver steps off a 5.0-m diving platform. Ignoring air resistance, what is the kinetic energy and velocity of the diver as she enters the water?
    • Known:
      • h_i = 5.0 m
      • h_f = 0 m
      • m = 68.2 kg
      • v_i = 0 m/s
    • Unknown:
      • KE_f = ?
      • v_f = ?
    • Solution:
      • The final GPE and the initial KE are 0 J.
      • Determine the initial GPE.
      • Use conservation of energy to determine the final KE.
      • Determine the final velocity.
      • PEi + KEi = PEf + KEf
      • mghi + 0 = 0 + KEf
      • KEf = mghi = (68.2 kg)(9.8 m/s^2)(5.0 m) = 3341.8 J
      • KEf = \frac{1}{2}mvf^2
      • vf = \sqrt{\frac{2KEf}{m}} = \sqrt{\frac{2(3341.8 J)}{68.2 kg}} = 9.9 m/s
    • The answer is reasonable for a diver entering the water.

Analyzing Collisions

  • Because the details of a collision can be very complex during the collision itself, the strategy is to find the motion of the objects just before and just after the collision.
  • In a closed, isolated system, momentum and energy are conserved.
  • Energy can change forms, so kinetic energy is not always conserved.

Types of Collisions

  • Elastic Collision: A collision in which the kinetic energy does not change.
    • Collisions between hard, elastic objects, such as those made of steel, glass, or hard plastic, often are called nearly elastic collisions.
  • Inelastic Collision: A collision in which kinetic energy decreases.
    • Objects made of soft, sticky material, such as clay, act in this way.
  • Perfectly Inelastic Collision: The objects stick together after colliding.
  • Superelastic Collision: The kinetic energy increases. If energy in the system is conserved, then one or more of the other forms of energy must decrease.

Example Problem: Ice Skaters

  • A 54.5-kg ice skater moving at 3.2 m/s collides with a 44.7-kg skater who is motionless. They then slide together along the frictionless ice. What is their velocity after the collision? How much kinetic energy was lost in the collision? What fraction of the original kinetic energy was lost?
    • Known:
      • m_A = 54.5 kg
      • v_{Ai} = 3.2 m/s
      • m_B = 44.7 kg
      • v_{Bi} = 0 m/s
    • Unknown:
      • v_f = ?
      • \Delta KE = ?
      • \frac{\Delta KE}{KE_i} = ?
    • Use conservation of momentum to determine the final velocity.
      • mA v{Ai} + mB v{Bi} = (mA + mB)v_f
      • vf = \frac{mA v{Ai} + mB v{Bi}}{mA + m_B} = \frac{(54.5 kg)(3.2 m/s) + (44.7 kg)(0 m/s)}{54.5 kg + 44.7 kg} = 1.8 m/s
    • Determine the initial and final kinetic energy.
      • KEi = \frac{1}{2} mA v{Ai}^2 + \frac{1}{2} mB v_{Bi}^2 = \frac{1}{2} (54.5 kg) (3.2 m/s)^2 + 0 = 278.72 J
      • KEf = \frac{1}{2} (mA + mB) vf^2 = \frac{1}{2} (54.5 kg + 44.7 kg) (1.8 m/s)^2 = 160.78 J
    • Calculate the change in energy and the fraction of energy lost.
      • \Delta KE = KEf - KEi = 160.78 J - 278.72 J = -117.94 J
      • \frac{\Delta KE}{KE_i} = \frac{-117.94 J}{278.72 J} = -0.43
    • A 43% loss of kinetic energy is reasonable.
    • Velocity is in meters per second and energy is in joules, so the units are correct.

Quiz Questions

  • The law of conservation of energy is defined for what type of system?
    • Correct Answer: C a closed, isolated system
  • What is the sum of the kinetic and the potential energy of the objects in a system called?
    • Correct Answer: B mechanical energy
  • As a skier goes down a slope, what happens to her kinetic energy (KE) and potential energy (PE)?
    • Correct Answer: D PE decreases, and KE increases.
  • In which type of collision does the kinetic energy remain constant?
    • Correct Answer: D elastic
  • In which type of collision do the objects stick together after colliding?
    • Correct Answer: B perfectly inelastic