Time–Speed–Distance (TSD) – UPSC CAPF Quantitative Aptitude Notes
For quick revision, practice & exam‑ready understanding.
1. Simple Explanation
The Time–Speed–Distance relationship is a fundamental concept in kinematics. It connects three basic variables:
| Variable | Symbol | Unit (SI) |
|---|---|---|
| Distance | (d) | kilometers (km) or meters (m) |
| Speed/Velocity | (v) | km / h or m / s |
| Time | (t) | hours (h) or seconds (s) |
Basic relation
[ \boxed{d = v \times t} ]
- Distance is the total ground covered.
- Speed is how fast the object moves (assumed constant).
- Time is how long the motion lasts.
If any one of these variables is missing, you can rearrange the formula:
[ t = \frac{d}{v} \qquad\text{or}\qquad v = \frac{d}{t} ]
2. Key Points to Remember
| # | Concept | Why it matters for CAPF |
|---|---|---|
| 1 | Assume constant speed unless specified otherwise. | All standard TSD problems use constant speed. |
| 2 | Units must match (e.g., km/h & h → km; m/s & s → m). | Unit mismatch => wrong answer. |
| 3 | Speed = distance / time | Often CAPF gives speed in km/h and wants distance in km. |
| 4 | Time = distance / speed | Useful when speed is given in m/s and time in minutes. |
| 5 | Distance = speed × time | Direct calculation if two variables are known. |
| 6 | Average speed: For varying speeds, divide total distance by total time. | Useful for “total distance covered in 3 hrs, where first 2 hrs at 40 km/h, next 1 hr at 60 km/h” type questions. |
| 7 | Speed conversions: 1 km/h = 1000 m / 3600 s ≈ 0.2778 m/s. | Convert when needed (e.g., from km/h to m/s). |
| 8 | Common traps: | |
| • Mixing up “speed” & “velocity” (direction irrelevant in TSD). | ||
| • Forgetting to divide by 60 or 3600 when converting minutes/hours. | ||
| • Using “speed” as distance (e.g., 60 km/h = 60 km in 1 hr). | ||
| 9 | Practice multiple-choice patterns: | |
| • “A train moves 120 km in 3 h. What is its speed?” | ||
| • “A cyclist covers 30 km in 1 h 15 min. Find speed.” | ||
| • “Two cars start from same point, one at 60 km/h, other at 40 km/h. After how many hours will the faster car be 80 km ahead?” |
3. Worked Example
Problem
A patrol vehicle travels 120 km in 3 hours.
- What is its average speed?
- If it maintains the same speed, how far will it travel in 5 hours?
- If the vehicle travels 50 km in 30 minutes, find its speed in km/h.
1. Average speed
[ v = \frac{d}{t} = \frac{120\text{ km}}{3\text{ h}} = 40\text{ km/h} ]
2. Distance in 5 hours
[ d' = v \times t' = 40\text{ km/h} \times 5\text{ h} = 200\text{ km} ]
3. Speed for 50 km in 30 min
First, convert time to hours: (30\text{ min} = 0.5\text{ h}).
[ v = \frac{50\text{ km}}{0.5\text{ h}} = 100\text{ km/h} ]
Answer Summary
| Question | Result |
|---|---|
| 1. Avg. speed | 40 km/h |
| 2. Distance in 5 h | 200 km |
| 3. Speed for 50 km in 30 min | 100 km/h |
4. Quick Revision Summary
| Concept | Formula | Quick Tip |
|---|---|---|
| Distance | (d = v \times t) | Multiply speed by time. |
| Speed | (v = \frac{d}{t}) | Divide distance by time. |
| Time | (t = \frac{d}{v}) | Divide distance by speed. |
| Unit check | km/h × h = km; m/s × s = m | Always verify units. |
| Average speed | (\frac{\text{Total distance}}{\text{Total time}}) | Add distances, add times. |
| Conversions | 1 km/h ≈ 0.2778 m/s | For meter‑second problems. |
| Common error | Mixing up 60 min = 1 h, 3600 s = 1 h | Remember 60 min = 1 h, 60 s = 1 min. |
Final Tips for UPSC CAPF
- Read carefully – Identify which variable is missing.
- Write down the given values with units.
- Choose the right formula – rearrange if necessary.
- Check units – convert if needed before calculation.
- Double‑check the answer for reasonableness (e.g., speed > 0, time > 0).
- Practice at least 3–5 problems a day to build speed.
Good luck with your preparation!