Core answer: the three kinematics equations — v=v₀+at, x=v₀t+½at², v²=v₀²+2ax; free fall is the special case v₀=0, a=g=9.8m/s²; projectile motion = independent horizontal uniform motion + vertical free fall; momentum p=mv is conserved in every collision, kinetic energy ½mv² only in elastic ones.
Five core formulas
| Scenario | Formula | Solves |
|---|---|---|
| Uniform acceleration | v=v₀+at | Final velocity |
| Displacement | x=v₀t+½at² | Any 3 of 4 variables |
| Time-free | v²=v₀²+2ax | When time is unknown |
| Free fall | h=½gt², v=√(2gh) | Height↔time↔speed |
| Momentum/Energy | p=mv, KE=½mv² | Collisions and energy |
Free-fall quick table
| Time | Height fallen | Impact speed |
|---|---|---|
| 1 s | 4.9 m | 9.8 m/s |
| 2 s | 19.6 m | 19.6 m/s |
| 3 s | 44.1 m | 29.4 m/s |
| 5 s | 122.5 m | 49 m/s |
| 10 s | 490 m | 98 m/s (air drag caps this) |
Height scales with time squared — double the time, quadruple the distance.
Decomposing projectile motion
The universal recipe: horizontal and vertical motions are independent. Horizontal: x = v₀·cosθ·t (constant speed). Vertical: y = v₀·sinθ·t − ½gt² (free-fall family). Maximum range comes at 45° in vacuum: R = v₀²·sin(2θ)/g. The [projectile calculator](/c/science/projectile) returns range, peak height and flight time from speed and angle.
Momentum vs kinetic energy
| Momentum p=mv | Kinetic energy ½mv² | |
|---|---|---|
| Type | Vector | Scalar |
| In collisions | Always conserved | Elastic only |
| Speed ×2 | ×2 | ×4 |
In an inelastic crash the lost kinetic energy becomes deformation, heat and sound — the physics behind crumple zones.
Example: how fast is a drop tower
A 40m free-fall section: impact v = √(2×9.8×40) = √784 = 28 m/s ≈ 100.8 km/h; time t = √(2h/g) ≈ 2.86 s. The [free-fall calculator](/c/science/free-fall) includes Moon (g=1.62) and other body presets.
Example: crash energy comparison
A 1.5t sedan at 60 vs 120 km/h: 60km/h (16.7m/s) → KE = ½×1500×16.7² ≈ 209 kJ; 120km/h (33.3m/s) → ≈ 832 kJ. Double the speed, quadruple the energy — 120km/h into a wall equals falling from a 56-meter building. "Speed kills" has a square law behind it.
Common mistakes
- "Heavier objects fall faster": in vacuum a feather and a hammer land together (Apollo 15 proved it on the Moon). Everyday differences are pure air resistance.
- "A projectile slows horizontally": ignoring drag, horizontal speed never changes — vertical speed grows, the combination draws the parabola.
- Mixing momentum and energy: "energy is conserved in collisions" is generally false (sound and heat leak it), but momentum always is — write the momentum equation first.
- Using g=10 for precision work: g varies 9.78-9.83 with latitude and altitude; use 9.8 for engineering and local values for precision.