Time – Speed – Distance (T‑S‑D)
Quantitative Aptitude – IIT‑JEE / Competitive Exam Level
1. Fundamental Relations
| Symbol | Meaning | Units |
|---|---|---|
| (S) | Distance travelled | km, m, km/h, m/s … |
| (T) | Time taken | h, min, s |
| (V) | Speed / Velocity | km/h, m/s |
| (a) | Acceleration | km/h², m/s² |
For uniform (constant) motion:
[ \boxed{S = V \times T} ]
[ \boxed{V = \frac{S}{T}} ]
[ \boxed{T = \frac{S}{V}} ]
Derivation
Distance = Speed × Time is equivalent to Speed = Distance / Time because speed is the rate of change of distance with respect to time.
2. Unit Consistency
| Conversion | Formula |
|---|---|
| (1\ \text{km} = 1000\ \text{m}) | (S_{\text{m}} = 1000 \times S_{\text{km}}) |
| (1\ \text{h} = 60\ \text{min} = 3600\ \text{s}) | (T_{\text{s}} = 3600 \times T_{\text{h}}) |
| (1\ \text{km/h} = \frac{1}{3.6}\ \text{m/s}) | (V_{\text{m/s}} = \frac{V_{\text{km/h}}}{3.6}) |
Tip: Convert all quantities to a common set of units before applying formulas.
3. Relative Motion (Two or More Objects)
3.1 Same Direction
For two objects moving in the same direction:
- Relative speed (V_{\text{rel}}) = (V_1 - V_2) (assuming (V_1 \ge V_2)).
- Distance between them after time (t): (D(t) = D_0 + (V_2 - V_1) t) where (D_0) is initial separation.
Time to catch up (when (V_1 > V_2)):
[ t = \frac{D_0}{V_1 - V_2} ]
3.2 Opposite Directions
For two objects moving in opposite directions:
- Relative speed (V_{\text{rel}}) = (V_1 + V_2).
- They meet when (t = \dfrac{D_0}{V_1 + V_2}).
4. Common Problem Types & Strategies
| Problem Type | Key Idea | Typical Formula |
|---|---|---|
| Single constant‑speed | Direct use of (S = V T) | (T = S/V) |
| Two trains / cars | Relative speed | (t = D_0/(V_1+V_2)) or (t = D_0/(V_1-V_2)) |
| Fastest/slowest | Compare speeds | (V_{\text{fast}} > V_{\text{slow}}) |
| Changing speeds | Break into segments | Sum of times / distances |
| Data sufficiency | Check if given data can determine quantity | Often need both (S) and (T) or (V) and (t) |
5. Tricky Points & Common Pitfalls
| Pitfall | Explanation | How to Avoid |
|---|---|---|
| Speed vs. velocity | In most aptitude problems speed = magnitude of velocity (positive scalar). | Treat speeds as positive numbers; ignore direction unless explicitly mentioned. |
| Units mix‑up | Using km/h with m and min simultaneously. | Convert all quantities to SI or all to metric units before calculation. |
| Assuming same starting point | “Two trains start from A and B” – they start at different points. | Identify the initial separation (D_0) correctly. |
| Relative speed in same direction | Using sum instead of difference. | Use (V_1 - V_2) (positive if the faster one catches up). |
| Time to meet vs. time to reach destination | Confusing the moment of meeting with arrival. | Draw a timeline or diagram. |
| Data sufficiency | Ignoring that one of the speeds is “average” over a variable schedule. | Check if the given speed is average or instantaneous. |
6. Derivation of Relative Speed Formula (Same Direction)
Let two objects move on a straight line with speeds (V_1, V_2) (same direction) and a fixed initial separation (D_0).
At time (t):
- Position of object 1: (x_1 = V_1 t)
- Position of object 2: (x_2 = D_0 + V_2 t)
They meet when (x_1 = x_2):
[ V_1 t = D_0 + V_2 t \implies (V_1 - V_2)t = D_0 \implies t = \frac{D_0}{V_1 - V_2} ]
Thus, (V_{\text{rel}} = V_1 - V_2).
7. Practice Question (Hard)
Question
Two buses, A and B, start from the same station S at 10:00 AM.
- Bus A travels at a constant speed of (60\ \text{km/h}).
- Bus B travels at a constant speed of (45\ \text{km/h}) but starts 15 minutes later.
- Bus B then accelerates uniformly to a speed of (75\ \text{km/h}) in the next (20) minutes and keeps that speed thereafter.
- Bus A reduces its speed by (20%) (i.e., to (48\ \text{km/h})) exactly at the instant when bus B reaches (75\ \text{km/h}).