Time–Speed–Distance (T‑S‑D) – Study Notes for Competitive Exams
(IIT‑JEE, JEE‑Main, IIT‑JEE Advanced, and other high‑stakes aptitude tests)
Why T‑S‑D matters
The Time–Speed–Distance (T‑S‑D) concept is the backbone of almost every word‑problem in quantitative aptitude. A solid grasp of the basic relationships, common traps, and the ability to manipulate units are indispensable for scoring high on the quantitative section.
1. The Three Fundamental Relations
| Quantity | Symbol | Formula | Units |
|---|---|---|---|
| Distance | s | s = v × t | km, m, mi, ft |
| Speed | v | v = s ÷ t | km h⁻¹, m s⁻¹ |
| Time | t | t = s ÷ v | h, min, s |
Key idea – All three variables are inter‑dependent. If any two are known, the third follows immediately.
2. Deriving the Core Equations
2.1 From Constant Velocity
If an object moves with constant speed v over a time interval t, the displacement s is
[ s = v \times t ]
Proof:
Velocity is defined as displacement per unit time: (v = \frac{s}{t}).
Rearrange → (s = v t).
2.2 For Variable Speed (Uniform Acceleration)
When speed changes uniformly (constant acceleration (a)), the kinematics equations are:
- (v = u + a t)
- (s = u t + \frac{1}{2} a t^2)
- (v^2 = u^2 + 2 a s)
Where
(u) – initial speed, (v) – final speed, (t) – time, (s) – distance.
These are derived from the definition of acceleration (a = \frac{\Delta v}{\Delta t}) and integrating velocity over time.
3. Common Pitfalls & Tricky Points
| Pitfall | What Happens | How to Avoid |
|---|---|---|
| Speed vs. Velocity | Confusing scalar vs. vector | Pay attention to direction in relative‑motion problems. |
| Unit Mixing | Mixing km/h with m/s | Convert all units before calculations. |
| Time in Different Units | Writing 30 min as 0.5 h but forgetting conversion | Use a conversion table (1 h = 60 min, 1 km = 1000 m). |
| Relative Speed | Adding speeds when moving away | If moving in opposite directions, relative speed = (v_1 + v_2); if same direction, relative speed = ( |
| Multiple Segments | Treating the entire journey as a single speed | Sum time or distance segment‑wise, then apply formulas. |
| Assuming Constant Speed | Ignoring acceleration/deceleration | Read the problem carefully; if “accelerates” or “decelerates” is mentioned, use kinematic equations. |
4. Relative Motion – The “Tricky” T‑S‑D
| Scenario | Relative Speed | Reason |
|---|---|---|
| Two trains moving towards each other | (v_1 + v_2) | Paths overlap; relative closing speed is the sum. |
| Two trains moving in the same direction | ( | v_1 - v_2 |
| A car and a cyclist moving in the same direction | ( | v_{\text{car}} - v_{\text{cyclist}} |
| A ship and a boat with currents | Add/subtract current speed | The current changes the effective speed relative to shore. |
Tip: Always draw a quick diagram or write the relative speed explicitly before computing time.
5. “Speed–Distance” with Multiple Stages (Piecewise)
If a journey consists of n stages, each with distance (s_i) and speed (v_i):
[ t_{\text{total}} = \sum_{i=1}^{n} \frac{s_i}{v_i} ]
Similarly, if n stages have known times (t_i) and speeds (v_i):
[ s_{\text{total}} = \sum_{i=1}^{n} v_i t_i ]
Remember: Always keep the unit consistency across all terms.
6. Useful Conversion Table
| Unit | Factor | Example |
|---|---|---|
| 1 h | 60 min | 1 h = 60 min |
| 1 h | 3600 s | 1 h = |