📚 Time‑Speed‑Distance Study Notes
(Quantitative Aptitude – Agniveer Program)
1️⃣ Simple Explanation
The Time‑Speed‑Distance (TSD) concept is the backbone of every motion‑based problem in aptitude tests.
It tells us how far an object travels when we know how fast it moves and for how long.
Formula
[ \text{Distance (S)} = \text{Speed (V)} \times \text{Time (T)} ]
- Speed (V) – how many units of distance are covered in one unit of time (e.g., km/hr, m/s).
- Time (T) – the duration of travel (e.g., hours, minutes).
- Distance (S) – the total length covered (e.g., km, meters).
Tip: Always keep the units consistent before plugging them into the formula.
2️⃣ Key Points & Common Variations
| Concept | What to Watch For | Typical Question Type |
|---|---|---|
| Basic TSD | (S = V \times T) | “A train travels 120 km at 60 km/h. How long?” |
| Speed Conversion | Convert km/h ↔ m/s, mph ↔ km/h, etc. | “A cyclist rides 30 km/hr for 2 hrs, find distance in meters.” |
| Average Speed | (\displaystyle \text{Avg V} = \frac{\text{Total Distance}}{\text{Total Time}}) | “A car covers 200 km in 4 hrs and 10 hrs for the return. Find average speed.” |
| Relative Motion | Use (V = V_1 + V_2) or (V_1 - V_2) depending on direction | “Two cyclists start from the same point in opposite directions. One goes 18 km/h, the other 12 km/h.” |
| Time & Distance relation | Rearrange (T = \frac{S}{V}) or (V = \frac{S}{T}) | “A bike travels 80 km in 4 hrs; find speed.” |
| Velocity & Direction | Positive/negative signs for direction; use vectors for 2‑D problems | “A boat moves 30 km upstream at 10 km/h, downstream at 20 km/h.” |
| Unit Consistency | Convert minutes to hours, miles to kilometers, etc. | “A bus travels 30 miles in 45 minutes.” |
Common Traps
| Trap | What’s Wrong | Fix |
|---|---|---|
| Mix‑up of units | Using km & h together, but later adding m | Convert all to a single unit system first. |
| Missing “average” | Using wrong speed in a round‑trip | Compute total distance and total time separately. |
| Direction confusion | Adding speeds when actually moving opposite | Use subtraction for opposite directions. |
| Time misreading | 2 hours 30 minutes → 2.5 hrs, not 2 hrs 30 min | Convert minutes to decimal hours (30 min = 0.5 hr). |
3️⃣ Worked Example
Problem
A cyclist travels 30 km in 1.5 hours.
Later, she rides the same distance back at a speed that is 10 km/h slower than her forward speed.
Find the total time taken for the round trip.
Step‑by‑Step
| Step | Calculation | Result |
|---|---|---|
| 1. Find forward speed | (V_f = \frac{S}{T} = \frac{30}{1.5} = 20 \text{ km/h}) | 20 km/h |
| 2. Compute return speed | (V_r = V_f - 10 = 20 - 10 = 10 \text{ km/h}) | 10 km/h |
| 3. Time for return leg | (T_r = \frac{S}{V_r} = \frac{30}{10} = 3 \text{ hrs}) | 3 hrs |
| 4. Total time | (T_{\text{total}} = T_f + T_r = 1.5 + 3 = 4.5 \text{ hrs}) | 4.5 hrs |
Answer: The round‑trip takes 4.5 hours.
4️⃣ Quick Revision Summary
| Concept | Formula | Quick Check |
|---|---|---|
| Distance | (S = V \times T) | Verify units: km/h × h = km |
| Speed | (V = \frac{S}{T}) | Check: distance ÷ time |
| Time | (T = \frac{S}{V}) | Check: distance ÷ speed |
| Average Speed | (\displaystyle \text{Avg V} = \frac{S_{\text{total}}}{T_{\text{total}}}) | Sum distances, sum times |
| Opposite Directions | (V_{\text{rel}} = V_1 + V_2) (if moving towards each other) | Add speeds |
| Same Direction | (V_{\text{rel}} = V_1 - V_2) (if one chasing) | Subtract speeds |
| Unit Conversion | 1 km = 1000 m, 1 hr = 60 min | Convert before calculation |
One‑Minute Cheat Sheet
- Speed unit → km/h, m/s, mph …
- Time unit → hours, minutes (min = 1/60 hr)
- Distance = Speed × Time
- If speed changes → compute each leg separately.
- Average speed = Total distance ÷ Total time.
Practice Tip:
Write down the given values, identify the unknown, and then decide which formula to apply.
Always check for “average” or “relative” speeds before plugging numbers.
Happy studying! 🚀