Short on time? This page is the recap. For explanations, bar charts and worked examples, use the full study guide. This is the algebra-based course: no calculus is needed.
Recap
- Energy is conserved in every interaction. It changes form inside a system or crosses the boundary as work.
- A single object can only have kinetic energy. Interacting objects (via gravity or springs) or a shape-changing object can also have potential energy.
- Mechanical energy E_mech = K + U.
- A change in one energy type is balanced by an equal and opposite change in other types, or by energy crossing the boundary.
- You can choose the system so that its total energy is constant. Putting Earth inside turns gravity’s work into ΔU_g.
- Friction and air resistance dissipate mechanical energy as thermal energy and sound.
Key relationships
| Relationship | When to use it |
|---|---|
| E_mech = K + U, with K = ½mv² | any system |
| ΔE_system = W_outside | always: energy in or out as work by outside forces |
| K_i + U_i = K_f + U_f | W_outside = 0 and no friction or air resistance inside |
| K_i + U_i = K_f + U_f + ΔE_thermal | W_outside = 0, friction inside the system |
| v = √(2gh) | from rest, smooth path, drop h (mass cancels) |
Factors of change: spring launcher, h ∝ Δx² (double the compression, four times the height). Smooth drop, v ∝ √h (four times the drop, twice the speed).
Assumptions behind the numbers
- “Smooth” or “frictionless” means no friction; air resistance is ignored unless stated.
- Springs are ideal and massless.
- Normal forces on a fixed track do no work because they act perpendicular to the motion.
- g = 9.8 m/s² at Earth’s surface. The course framework says exam questions use 10 m/s², and 9.8 m/s² is also accepted.
Mistakes to avoid
- No system named. “Potential energy turns into kinetic energy” needs the object + Earth system.
- Counting gravity twice: using both ΔU_g and the work done by gravity for the same system.
- Changing the zero part-way through a problem.
- Saying energy is “lost” with friction without saying where it went (thermal energy, sound).
- Forgetting a term, such as U_g as a spring-launched ball rises off the spring.
- Expecting mass to matter on a smooth slope.
- Doubling instead of quadrupling when a spring’s compression doubles.
Quick self-check
- A stone is dropped from rest and falls 5.0 m. Ignore air resistance. How fast is it moving? (9.9 m/s)
- A 0.50 kg block on a smooth floor is pushed against a spring (k = 200 N/m), compressing it 0.10 m, then released. How fast does it leave the spring? (2.0 m/s)
- A system has 100 J of mechanical energy at the start and 70 J at the end. No outside force did work. Where did 30 J go? (Into thermal energy and sound, through friction or air resistance inside the system.)
- System = ball only. Does the ball’s energy stay constant as it falls? (No. Gravity is an outside force doing positive work, so K increases.)
Next: practice questions.