Motion Word Problems
Master the Distance-Rate-Time Relationship for SAT Success
📐 Section 1: The Fundamental Relationship
The Master Formula
$$d = r \times t$$
Distance = Rate × Time
Understanding Rate: Rate is speed — the distance traveled per unit of time. If a car travels at 60 mph, it covers 60 miles every hour at a constant (steady) speed.
Three Forms of the Formula
| Find What? | Formula | Example |
|---|---|---|
| Distance | $d = r \times t$ | $d = 60 \times 3 = 180$ miles |
| Rate | $r = \dfrac{d}{t}$ | $r = \dfrac{180}{3} = 60$ mph |
| Time | $t = \dfrac{d}{r}$ | $t = \dfrac{180}{60} = 3$ hours |
⚠️ Units Must Match!
- If rate is in miles per hour, time must be in hours
- If rate is in meters per second, time must be in seconds
- Always convert units before calculating!
Common Time Conversions
| 30 minutes | = 0.5 hours |
| 45 minutes | = 0.75 hours |
| 20 minutes | = $\frac{1}{3}$ hour |
| 40 minutes | = $\frac{2}{3}$ hour |
| 1 hour | = 60 minutes = 3600 seconds |
Speed Conversions
- mph → ft/s: multiply by $\frac{22}{15}$
- km/h → m/s: divide by 3.6
🛠️ Section 2: Setting Up Motion Problems — The 5-Step Process
- Read carefully — Identify all given information. What is being asked? Note units.
- Define variables — Clearly state what each variable represents (include units).
- Draw a diagram & create a table — Sketch the scenario. Organize info in a Rate-Time-Distance table.
- Write equations — Use $d = rt$ for each object. Use relationships between objects.
- Solve, check, and answer — Solve the equation. Verify. Answer what was asked with units.
🔑 Key Relationships
| Same distance | $d_1 = d_2$ |
| Same time | $t_1 = t_2$ |
| Total distance | $d_1 + d_2 = D_{\text{total}}$ |
| Total time | $t_1 + t_2 = T_{\text{total}}$ |
| Head start (time) | $t_{\text{slow}} = t_{\text{fast}} + \Delta t$ |
📋 The Table Method
The most powerful organizational tool for motion problems:
| Object | Rate (r) | Time (t) | Distance (d = r·t) |
|---|---|---|---|
| Car A | 50 mph | $t$ | $50t$ |
| Car B | 70 mph | $t - 1$ | $70(t-1)$ |
Fill in known values first, then use variables for unknowns. The Distance column is always Rate × Time.