Short on time? This page is the recap for the calculus-based course (separate from Physics 1). For explanations, a graph and worked examples, use the full study guide.
Recap
- Translational kinetic energy is the energy of motion from place to place: K = ½mv². For a system, use the speed of its centre of mass.
- Unit: the joule, 1 J = 1 kg·m²/s². Mass in kg, speed in m/s.
- K is a scalar. It depends on speed only, never on direction. K ≥ 0; ΔK can have either sign.
- K depends on the reference frame: observers moving relative to each other measure different speeds, so different K.
Key relationships
| Relationship | Meaning |
|---|---|
| K = ½mv² | definition; v is speed |
| v² = v_x² + v_y² + v_z² | square components, then add |
| v = √(2K/m) | speed from K |
| K ∝ m (fixed v); K ∝ v² (fixed m) | factor-of-change rules |
| dK/dt = m v_x a_x (1D); m v · a in general | sign tells whether K rises or falls |
| K′ = ½m|v − u|² | K seen from a frame moving at u |
Graph shapes. K–v: parabola through the origin. K–v²: straight line, slope ½m. K–m at fixed v: straight line, slope ½v². K–t for a projectile: U-shape with a minimum of ½mv_x² at the top, not zero.
Assumptions behind the numbers
- The object is modelled as a point (or a system’s centre of mass); rotation is ignored until Unit 6.
- One inertial frame for the whole problem unless the question compares frames.
- Free fall: a_y = −g with g = 9.8 m/s² (+y upward), air resistance ignored.
Mistakes to avoid
- K with a sign or direction. It has neither.
- Doubling v doubles K. It quadruples K.
- ΔK = ½m(Δv)². Compute K_f − K_i.
- Adding velocity components as numbers. Use v² = v_x² + v_y².
- K = 0 at the top of a projectile path. K_top = ½mv_x².
- Mass in grams. Convert to kg.
- Mixing frames. Measure every speed relative to the same observer.
Quick self-check
- What is K for a 0.060 kg ball at 25 m/s? (½ × 0.060 × 625 = 19 J; 18.75 J)
- A 2.0 kg object has v = (3.0, −4.0, 12) m/s. Find K. (v² = 9 + 16 + 144 = 169 m²/s², so K = 169 J ≈ 170 J)
- K of a cart is multiplied by 4 at the same mass. By what factor did its speed change? (× 2)
- A 1.5 kg cart has v_x = +4.0 m/s and a_x = −2.0 m/s². Find dK/dt. (1.5 × 4.0 × (−2.0) = −12 J/s; K is falling)
Next: practice questions.