Time–Speed–Distance (TSD) – Study Notes
(For Maharashtra Patwari / Lekhpal – Quantitative Aptitude)
Goal – Master the fundamentals, derivations, and pitfalls of the TSD relationship so you can crack any quantitative problem in the exam.
1. Basic Definitions
| Symbol | Meaning | Unit (SI) | Typical Use |
|---|---|---|---|
| (s) | Distance travelled | metres (m) | Total ground covered |
| (u) | Initial speed | m s⁻¹ | Speed at (t = 0) |
| (v) | Final speed | m s⁻¹ | Speed at time (t) |
| (t) | Time taken | seconds (s) | Duration of motion |
| (a) | (Optional) Acceleration | m s⁻² | Rate of change of speed |
Key Idea:
( \displaystyle s = \text{(average speed)} \times t )
For constant speed, “average speed” = that constant speed.
2. Deriving the Core Formula
2.1 Constant Speed
If speed stays unchanged, the distance covered in a small time interval (dt) is (v,dt).
Integrating over the whole duration (t):
[ s = \int_{0}^{t} v,dt = v , t ]
2.2 Variable Speed (Uniform Acceleration)
When acceleration (a) is constant, speed varies linearly:
(v = u + a t).
Distance is the integral of speed:
[ s = \int_{0}^{t} (u + a t'),dt' = u t + \frac{1}{2} a t^2 ]
Takeaway – The simple s = vt holds only when speed is constant; otherwise use the full expression.
3. Average Speed – Two Special Cases
| Situation | Average Speed Formula | Explanation |
|---|---|---|
| Equal time intervals | (\displaystyle \bar v = \frac{v_1 + v_2}{2}) | Arithmetic mean of speeds |
| Equal distance intervals | (\displaystyle \bar v = \frac{2 v_1 v_2}{v_1 + v_2}) | Harmonic mean of speeds |
Why the difference?
• When equal times are taken, the distance travelled in each part is proportional to the speed → arithmetic mean.
• When equal distances are taken, the time spent at each speed is inversely proportional to that speed → harmonic mean.
4. Relative Speed
Used when two objects move towards or away from each other.
| Direction | Relative speed |
|---|---|
| Moving towards | (v_{\text{rel}} = v_1 + v_2) |
| Moving apart | (v_{\text{rel}} = |
Example – Two cars moving towards each other at 40 km/h and 60 km/h will cover the distance between them at 100 km/h.
5. Unit Conversions – A Must‑Know Trick
| Common unit | Conversion to m s⁻¹ |
|---|---|
| km/h | ( \displaystyle \frac{1000}{3600} = 0.27778) |
| m/min | ( \displaystyle \frac{1}{60}) |
| km/min | ( \displaystyle \frac{1000}{60}) |
| km/s | ( \displaystyle 1000) |
Remember – Always convert to a common unit before plugging into formulas.
6. Common Pitfalls & Tricky Points
| Pitfall | Why it Happens | Fix |
|---|---|---|
| Mixing units | 1 km/h ≠ 1 m/s | Convert all speeds to the same unit first |
| Assuming “average speed” = arithmetic mean | Only valid for equal times | Use harmonic mean if distances are equal |
| Neglecting direction | Speed is scalar, but distance/relative speed can be signed | Keep track of “towards” / “away” |
| Overlooking acceleration | Problem states “constant speed” → no acceleration term | Use (s = vt) only if told “speed is constant” |
| Adding speeds when directions differ | Add magnitudes, then assign sign | (v_{\text{rel}} = v_1 - v_2) if opposite directions |
7. Quick‑Reference Formula Sheet
| Scenario | Formula |
|---|---|
| Distance with constant speed | (s = vt) |
| Distance with constant acceleration | (s = ut + \frac{1}{2} a t^2) |
| Average speed (equal times) | (\bar v = \frac{v_1+v_2}{2}) |
| Average speed (equal distances) | (\bar v = \frac{2v_1v_2}{v_1+v_2}) |
| Relative speed (towards) | (v_{\text{rel}} = v_1 + v_2) |
| Relative speed (apart) | (v_{\text{rel}} = |
| Time to cover distance | (t = \frac{s}{v}) |