MOTION BASICS
Always define your + direction first. All signs follow from that choice.
displacement
Δd = d_f − d_i
speed
scalar: distance / time
Distance = total path (scalar)
Displacement = net change in position (vector)
Speed uses distance · velocity uses displacement
VARIABLE KEY
| Symbol | Meaning | Unit |
|---|
| d | displacement | m |
| v₀ | initial velocity | m/s |
| v | final velocity | m/s |
| a | acceleration | m/s² |
| t | time | s |
| g | gravity | 9.8 m/s² |
UAM EQUATIONS (use only when a = constant)
Pick the equation that has your 3 known variables + 1 unknown. Eq 3 has no t — use when t is missing.
g = 9.8 m/s² downward
If up is +: use a = −9.8 m/s²
If down is +: use a = +9.8 m/s²
CHOOSING AN EQUATION
| Missing variable | Best equation |
|---|
| t is unknown | Eq 3 (v² = v₀²+2ad) |
| v is unknown | Eq 2 (d = v₀t+½at²) |
| a is unknown | Eq 4 (d = avg·v·t) |
| d is unknown | Eq 1 (v = v₀+at) |
MOTION GRAPHS
| Graph | Slope = | Area = |
|---|
| d-t | velocity | — |
| v-t | acceleration | displacement |
| a-t | jerk (rare) | Δv |
- Straight d-t line → constant velocity
- Curved d-t line → accelerating
- Horizontal v-t line → constant velocity (a = 0)
- Slope down on v-t → deceleration
- Area below x-axis on v-t → negative displacement
PROJECTILE MOTION
Horizontal and vertical are independent. Time links them.
aₓ
0 (no horizontal force)
At peak height: vᵧ = 0 (but vₓ remains unchanged)
COMMON MISTAKES
- Forgetting direction sign (+/−)
- Using total distance instead of displacement for velocity
- Mixing horizontal and vertical in projectile problems
- Using UAM equations when a is NOT constant
- Wrong sign for g (check: is up your + direction?)
- Forgetting to find time before finding range in projectile
QUICK EXAMPLE — FREE FALL
Ball dropped from rest, falls 20 m:
v² = v₀² + 2ad
v² = 0 + 2(9.8)(20) = 392
v = √392 ≈ 19.8 m/s
v = v₀ + at
t = v/a = 19.8/9.8 ≈ 2.02 s
QUICK EXAMPLE — PROJECTILE
Horizontal launch at 12 m/s from 20 m cliff:
Vertical: 20 = ½(9.8)t² → t ≈ 2.02 s
Horizontal: x = 12 × 2.02 ≈ 24.2 m
PROJECTILE STRATEGY
- Split v₀ into vₓ = v₀cosθ and vᵧ₀ = v₀sinθ
- Use vertical equations to find time of flight
- Use t to find horizontal range: x = vₓ·t
- At peak: vᵧ = 0 → use to find max height