9.1 Forms and Conservation of Energy

9.1 Forms and Conservation of Energy

Quick Answer: Kinetic energy (KE) depends on mass and the square of speed ((KE = \frac{1}{2}mv^2)), so doubling speed multiplies KE by four. Potential energy (PE) is stored energy (gravitational, elastic, chemical). In a closed system without non-conservative work, mechanical energy is conserved—it transforms among forms (chemical → electrical → mechanical; gravitational PE ↔ KE on a roller coaster) but the total remains constant. Friction on an incline converts mechanical energy into thermal energy, so KE + PE decreases even though total energy of system + surroundings is conserved.

Energy questions on Praxis Middle School Science (5442) sit in Physical Science (ETS II.C) and reward precise reasoning: which quantity changes, which form appears, and what conservation forbids. Treat every scenario as an energy inventory—list what enters, what leaves, and what changes form.

Kinetic energy: mass and speed

Kinetic energy is energy of motion. The defining relationship is:

[ KE = \frac{1}{2}mv^2 ]

where (m) is mass and (v) is speed. Two exam-critical implications follow immediately:

1. Mass is linear. Doubling mass at the same speed doubles KE.
2. Speed is squared. Doubling speed at the same mass quadruples KE (factor of (2^2 = 4)). Tripling speed multiplies KE by nine.

Classroom check: A 2 kg cart at 3 m/s has (KE = \frac{1}{2}(2)(9) = 9) J. The same cart at 6 m/s has (KE = \frac{1}{2}(2)(36) = 36) J—four times larger, not twice. Exam traps often offer “twice the energy” when speed doubles; choose the factor-of-four answer.

Potential energy and major energy forms

Potential energy is stored energy associated with position or configuration. Common middle-school forms include:

• Gravitational PE — energy due to height in a gravitational field ((PE_g = mgh) near Earth’s surface). Raising an object stores energy; lowering releases it as KE (or work/heat if dissipated).
• Elastic (spring) PE — energy stored in a stretched or compressed spring ((PE_s = \frac{1}{2}kx^2)). Greater displacement from equilibrium stores more energy.
• Chemical PE — energy in chemical bonds (batteries, food, fuels). Released during reactions that produce more stable products.
• Electrical energy — energy associated with electric charge and circuits; often a middle form between chemical storage and mechanical output (motors).
• Thermal energy — microscopic kinetic energy of particles; increases with temperature for a given substance.
• Sound energy — organized mechanical vibration of a medium (air, water, solids).
• Light (radiant) energy — electromagnetic energy; can be absorbed, reflected, or transmitted by matter.

Conservation of energy

The law of conservation of energy states that energy cannot be created or destroyed—only transferred or transformed. In a closed system with no external work or heat exchange, the total energy remains constant. For mechanical systems without friction:

[ ME = KE + PE = \text{constant} ]

Classic illustrations:

Pendulum

At the highest points, speed is briefly zero: max PE, min KE. At the bottom, height is lowest: max KE, min PE. Idealized swings exchange PE and KE with constant total mechanical energy. Real pendulums slowly lose amplitude because air resistance and pivot friction convert mechanical energy into thermal energy of the air and hardware—total energy of system + surroundings still balances.

Vertical spring–mass

Compress a spring and release: elastic PE converts to KE as the mass moves through equilibrium, then to PE (elastic and/or gravitational, depending on orientation) at the opposite extreme. Again, frictionless models conserve mechanical energy; real systems warm slightly.

Roller coaster

At the top of the first hill (assuming negligible friction), cars have mostly gravitational PE. Descending, PE decreases while KE increases. At the bottom, KE is greatest. Climbing the next hill reverses the exchange. Designers compare heights: a later hill cannot exceed the first without an energy input (lift motors), because conservation forbids spontaneous gain of mechanical energy.

Energy transformations students must track

Praxis items often narrate a chain and ask which statement is true or which form is missing:

1. Chemical → electrical → mechanical: A battery stores chemical PE. In a circuit, chemical energy becomes electrical energy. A motor converts electrical energy into mechanical KE (plus waste heat).
2. Chemical → thermal → mechanical: Burning fuel heats gas that expands and moves a piston.
3. Electrical → light + thermal: An incandescent bulb radiates visible light and substantial heat; LEDs are more efficient at light but still produce some thermal energy.
4. Mechanical → sound + thermal: Clapping hands or scraping brakes.

When analyzing a device, ask: Where was energy stored initially? What useful form leaves? What waste form appears? Waste heat is not “destroyed energy”—it is energy transferred to the surroundings.

Friction on an incline: mechanical energy decreases

Imagine a block sliding down a rough ramp. Gravity does work that would increase KE, but kinetic friction does negative work on the block, converting mechanical energy into thermal energy of the block and ramp surfaces.

Energy bookkeeping for the block + ramp + Earth system:

• Gravitational PE decreases as height drops.
• KE may increase, but by less than the PE loss if friction is present.
• The “missing” mechanical energy appears as increased thermal energy (surfaces feel warmer).

Exam stem pattern: “Compared with a frictionless ramp of the same height, the block’s speed at the bottom is…” → smaller, because some PE became thermal rather than KE. Another pattern: “Which quantity decreases for the sliding block considered alone?” → mechanical energy (KE + PE), while total energy including thermal remains conserved if the thermal energy of the surfaces is counted.

Collisions and the “both at rest” impossibility

In a collision between two objects, linear momentum of an isolated system is conserved, and energy accounting depends on whether the collision is elastic or inelastic. A high-yield conceptual trap: two free objects moving toward each other cannot both be at rest after colliding unless an external agent removes momentum and energy.

Why? If both end at rest, total momentum becomes zero. That is possible only if the initial total momentum was already zero (equal and opposite momenta). Even then, kinetic energy cannot simply vanish without becoming another form (deformation, heat, sound). Two identical masses with equal speeds toward each other can stop in a perfectly inelastic “stick and drop” only if something else (like locking into a massive support) absorbs momentum—otherwise they move together with zero total momentum but may still have deformation energy. For free isolated particles, “both at rest afterward” with nonzero initial KE violates conservation unless that KE is accounted for elsewhere and momentum already summed to zero.

Safer exam takeaway: You cannot create or destroy energy. If KE disappears from the macroscopic motion, look for deformation, heat, or sound. If both objects are claimed to stop with no other energy form identified, the description is incomplete or impossible for an isolated system with nonzero initial mechanical energy and inconsistent momentum.

Teaching and item strategies

• Translate story problems into before/after energy tables.
• Watch for speed-squared distractors on KE.
• On coasters and pendulums, match extreme positions to max/min KE and PE.
• On devices, follow the transformation chain and name waste heat.
• On friction, state clearly that mechanical energy falls while total energy (including thermal) is conserved.

Master these patterns and II.C energy items become inventory problems rather than memorization contests.