Short on time? This page is the recap for the calculus-based course (separate from Physics 1). For the graphs, bar charts and worked examples, use the full study guide.
Recap
- Choose the system before writing any energy term. Everything outside is the surroundings.
- A single object can only have kinetic energy. A system with conservative interactions, or one that deforms reversibly, can also have potential energy.
- Mechanical energy E = K + U.
- Energy is conserved in every interaction. A change in one type inside the system is balanced by changes in other types or by energy crossing the boundary.
- Nonconservative forces such as kinetic friction dissipate mechanical energy as thermal energy and sound.
Key relationships
| Relationship | Units | Use it when |
|---|---|---|
| E_mech = K + U | J | always (definition) |
| ΔK + ΔU + ΔE_th = W_ext | J | any system |
| K_i + U_i = K_f + U_f | J | W_ext = 0 and no friction inside |
| v(x) = √[2(E − U(x))/m] | m/s | object in a U(x) system |
| turning point: U(x) = E | — | K = 0 there |
| v_esc = √(2GM/R) | m/s | U_g = −GMm/r, from the surface |
| ΔE_th = ∫ f_k ds (= f_k d if constant) | J | sliding friction |
Assumptions behind the numbers
- Ideal springs, frictionless surfaces and no air resistance unless stated.
- A planet or moon is treated as fixed when the other object is much lighter.
- g = 9.8 m/s²; G = 6.67 × 10⁻¹¹ N·m²/kg².
Mistakes to avoid
- Potential energy for a one-object system. Include Earth or use work done by gravity, not both.
- “Friction breaks energy conservation.” Total energy is conserved; mechanical energy is not.
- Constant speed means constant energy. U can still change while an external force does work.
- Turning points at the top of U(x). They are where U = E.
- mgh for large heights. Use −GMm/r.
- f_k d when μ_k varies. Integrate.
Quick self-check
- A ball is dropped from rest and falls 0.45 m with no air resistance. Speed? (√(2 × 9.8 × 0.45) = 3.0 m/s)
- A 0.20 kg object has E = 1.0 J and is at a point where U = 0.64 J. Speed? (√(2 × 0.36 ÷ 0.20) = 1.9 m/s)
- A sledge’s K falls from 50 J to 38 J on a level patch. What happened to the 12 J? (Dissipated as thermal energy and sound by friction.)
Next: practice questions.