Time, Speed and Distance — Study Notes

Overview

Time, Speed and Distance is a high-yield topic in SSC CGL Tier-1, appearing in 2–4 questions per exam. The topic tests your ability to manipulate the fundamental relationship **Distance = Speed × Time** and apply it to real-world scenarios involving trains crossing objects, boats moving in rivers, and relative motion between two moving bodies.

Mastery requires fluency in unit conversions (especially km/h ↔ m/s), recognizing problem patterns, and executing calculations quickly. Most questions involve either trains (crossing platforms, poles, or other trains), boats and streams (upstream/downstream speed adjustments), or relative speed (two objects moving toward or away from each other). Average speed problems test weighted averaging when speeds change across segments.

Strong command of this topic directly translates to marks because the problems follow predictable templates. Focus on speed of calculation and recognizing which formula applies instantly.

Key Concepts

• **Fundamental relationship**: Distance = Speed × Time. Any one quantity can be found if the other two are known. Rearrange as Speed = Distance/Time or Time = Distance/Speed as needed.

• **Unit conversion**: 1 km/h = 5/18 m/s. To convert km/h to m/s, multiply by 5/18. To convert m/s to km/h, multiply by 18/5. This is critical for train problems where lengths are in meters and speeds in km/h.

• **Relative speed — same direction**: When two objects move in the same direction, their relative speed is the difference of their speeds (Speed₁ − Speed₂). Used when one train overtakes another.

• **Relative speed — opposite direction**: When two objects move toward each other, their relative speed is the sum of their speeds (Speed₁ + Speed₂). Used when two trains cross each other head-on.

• **Train crossing problems**: When a train crosses a stationary object (pole, man), the distance covered equals the length of the train. When crossing a platform or bridge, distance = length of train + length of platform/bridge.

• **Boats and streams**: Let speed of boat in still water = b km/h and speed of stream = s km/h. Then upstream speed = (b − s) km/h and downstream speed = (b + s) km/h. The stream opposes upstream motion and aids downstream motion.

• **Average speed**: Average speed is **not** the arithmetic mean of speeds. It equals Total Distance / Total Time. For equal distances at two speeds, use the harmonic mean formula: Average Speed = (2 × Speed₁ × Speed₂) / (Speed₁ + Speed₂).

• **Meeting and crossing time**: If two objects start from opposite ends and meet, time to meet = Total Distance / (Speed₁ + Speed₂). If they start from the same point moving in opposite directions, they separate at relative speed (Speed₁ + Speed₂).

Formulas / Key Facts

1. **Distance = Speed × Time** — Core relationship for all problems.

2. **Speed in m/s = Speed in km/h × (5/18)** — Standard unit conversion.

3. **Speed in km/h = Speed in m/s × (18/5)** — Reverse conversion.

4. **Relative speed (opposite directions) = Speed₁ + Speed₂** — Trains meeting head-on, objects approaching.

5. **Relative speed (same direction) = |Speed₁ − Speed₂|** — One train overtaking another.

6. **Time for train to cross pole/man = (Length of train) / (Speed of train)** — Distance equals train length only.

7. **Time for train to cross platform = (Length of train + Length of platform) / (Speed of train)** — Total distance covered.

8. **Upstream speed = Boat speed − Stream speed (b − s)** — Moving against the current.

9. **Downstream speed = Boat speed + Stream speed (b + s)** — Moving with the current.

10. **Boat speed in still water = (Downstream speed + Upstream speed) / 2** — Average of the two.

11. **Stream speed = (Downstream speed − Upstream speed) / 2** — Half the difference.

12. **Average speed for equal distances = (2 × Speed₁ × Speed₂) / (Speed₁ + Speed₂)** — Harmonic mean formula.

Worked Examples

**Example 1: Train crossing a pole** A train 150 m long passes a pole in 15 seconds. Find its speed in km/h.

**Solution:** Distance = Length of train = 150 m Time = 15 s Speed = Distance / Time = 150 / 15 = 10 m/s Convert to km/h: 10 × (18/5) = 36 km/h **Answer: 36 km/h**

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**Example 2: Two trains crossing each other** Train A (length 120 m, speed 54 km/h) and Train B (length 180 m, speed 72 km/h) are moving in opposite directions. How long will they take to cross each other?

**Solution:** Total distance to cover = 120 + 180 = 300 m Speed of Train A = 54 km/h = 54 × (5/18) = 15 m/s Speed of Train B = 72 km/h = 72 × (5/18) = 20 m/s Relative speed = 15 + 20 = 35 m/s (opposite directions) Time = Distance / Relative Speed = 300 / 35 = 60/7 ≈ 8.57 seconds **Answer: 60/7 seconds or approximately 8.57 seconds**

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**Example 3: Boat and stream** A boat travels 24 km downstream in 2 hours and returns upstream in 3 hours. Find the speed of the boat in still water and the speed of the stream.

**Solution:** Downstream speed = 24 / 2 = 12 km/h Upstream speed = 24 / 3 = 8 km/h Boat speed in still water = (12 + 8) / 2 = 10 km/h Stream speed = (12 − 8) / 2 = 2 km/h **Answer: Boat = 10 km/h, Stream = 2 km/h**

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**Example 4: Average speed** A person travels from A to B at 40 km/h and returns from B to A at 60 km/h. What is the average speed for the entire journey?

**Solution:** Use harmonic mean formula (equal distances): Average speed = (2 × 40 × 60) / (40 + 60) = 4800 / 100 = 48 km/h **Answer: 48 km/h**

Common Mistakes

1. **Adding speeds to find average speed** → Incorrect. Average speed = Total Distance / Total Time, not (Speed₁ + Speed₂) / 2. Use the harmonic mean formula for equal distances.

2. **Forgetting unit conversion** → Train lengths are usually in meters, speeds in km/h. Convert speed to m/s (multiply by 5/18) before calculating time in seconds.

3. **Using train length only when crossing a platform** → When a train crosses a platform or bridge, total distance = train length + platform length. For a pole or standing person, use train length only.

4. **Confusing upstream and downstream** → Downstream is faster (boat + stream), upstream is slower (boat − stream). Always subtract stream speed from boat speed for upstream.

5. **Wrong relative speed direction** → If two objects move in the same direction, relative speed is the **difference** (subtract). If opposite directions (meeting/crossing), relative speed is the **sum** (add).

Quick Reference

• **Core formula**: Distance = Speed × Time. Rearrange as needed.
• **Conversion**: km/h to m/s → multiply by 5/18; m/s to km/h → multiply by 18/5.
• **Train crossing pole**: Time = Train length / Speed.
• **Train crossing platform**: Time = (Train length + Platform length) / Speed.
• **Relative speed (opposite)**: Speed₁ + Speed₂.
• **Relative speed (same direction)**: Speed₁ − Speed₂.
• **Downstream**: Boat speed + Stream speed.
• **Upstream**: Boat speed − Stream speed.
• **Average speed (equal distances)**: 2ab / (a + b), where a and b are the two speeds.