Eleventh Grade · Science · week 17

Conservation of Mechanical Energy and Power

Apply conservation of mechanical energy to a frictionless system and compute power as the rate of doing work.

When only gravity (or a spring) does work on an object -- no friction or other energy-draining force involved -- the total mechanical energy, KE + PE, stays exactly constant throughout the motion. Energy is not created or destroyed; it simply converts back and forth between kinetic and potential forms. A ball rolling down a frictionless ramp trades potential energy for kinetic energy in exact balance; a pendulum swinging without air resistance trades the same two forms back and forth forever, always reaching the same height on each side. This single idea, KE_initial + PE_initial = KE_final + PE_final, often solves problems far faster than tracking forces and accelerations the whole way through.

Power is a related but different idea: it measures how QUICKLY energy is transferred or work is done, P = W/t, measured in watts (W), where 1 watt = 1 joule per second. Two machines can do the exact same amount of work, but the one that does it faster is more powerful. When a force is constant and aligned with a constant velocity, power can also be written as P = F*v -- useful for problems described in terms of speed rather than time or distance.

📋 For the teacher
Teach it (10–15 min):
  • This week's lab: build a simple curved track from a foil-covered pool noodle, a flexible ruler, or a length of foam pipe insulation cut lengthwise, and roll a marble down one side and partway up the other. It should rise close to (but a little less than -- friction is never quite zero) its starting height, a visible demonstration of energy conservation.
  • Point out the striking result baked into the worked example: mass cancels out of v=sqrt(2*g*h) completely, meaning a bowling ball and a marble released from the same frictionless height reach the bottom at the same speed -- the same surprising fact from free fall, now showing up again through an entirely different derivation.
  • Compare two ways of climbing the same flight of stairs -- walking slowly versus running up -- to make the power concept concrete: the WORK done against gravity is identical either way, but the faster climb clearly demands more power.
  • When a problem gives a height and asks for a final speed with no mention of time, treat that as a strong hint to reach for conservation of energy rather than the kinematic equations from weeks 4-6.
Talk about it:
  • Why does an object's mass not affect its final speed when sliding down a frictionless ramp from a given height?
  • What is the difference between work and power?
  • How would this week's worked example change if there were significant friction on the ramp?

Mastery looks like: They apply conservation of mechanical energy to find an unknown speed or height in a frictionless system, and correctly compute power from work and time (or force and velocity).

Common stumbles: Forgetting that conservation of energy in this simple form only applies when friction (or another energy-draining force) is absent or ignored; confusing work (energy transferred) with power (the rate of that transfer).

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