8. Speed, Time and Distance Shortcuts

Why This Chapter Matters

This chapter rewards clean unit conversion and calm setup. It covers straight-line motion, average speed, relative speed, trains, platform crossing, and time loss or gain from speed changes.

Core Ideas

• Use $\text{distance}=\text{speed}\times \text{time}$ as the base relation.
• Convert $\text{m/s}$ to $\text{km/h}$ by multiplying by $\frac{18}{5}$.
• When a train crosses a pole, only the train length matters. When it crosses a platform or another train, add lengths.
• Relative speed handles chase and opposite-direction questions neatly.
• For equal distances travelled at two different speeds, the average speed is the harmonic mean, not the simple mean.
• If speed changes but distance remains fixed, time changes inversely.

High-Value Formulas

How To Approach Questions

1. Convert all speeds into one unit before computing.
2. Write the effective distance that must be covered.
3. Use relative speed for moving-object interactions.
4. For return journeys or equal-distance problems, choose the right average-speed formula instead of averaging directly.
5. If the question gives total journey time with different segment speeds, build the equation from the segment times.

Worked Examples

Example 1

Prompt: A train $180$ metres long crosses a pole in $9$ seconds. Find its speed.

Approach: Speed $=180÷9=20\text{ m/s}=72\text{ km/h}$.

Example 2

Prompt: At $72\text{ km/h}$, how much distance is covered in $25$ seconds?

Approach: Convert speed to $20\text{ m/s}$. Distance $=20\times 25=500\text{ m}$.

Example 3

Prompt: A traveller covers a distance at $40\text{ km/h}$ and returns over the same distance at $10\text{ km/h}$. Find the average speed for the whole trip.

Approach: For equal distances, average speed $=\frac{2ab}{a+b}=\frac{2\times 40\times 10}{40+10}=16\text{ km/h}$.

Example 4

Prompt: A car covers a distance in $10$ hours, moving at $40\text{ km/h}$ for the first half of the time and $20\text{ km/h}$ for the second half. Find the distance.

Approach: The first $5$ hours cover $200$ km and the next $5$ hours cover $100$ km, so the total distance is $300$ km.

Common Mistakes

• Mixing metres with kilometres or hours with seconds.
• Using train length alone when a platform is also being crossed.
• Missing the sign change in relative speed questions.
• Taking a simple average of speeds when the distances are equal.
• Forgetting that speed gain and time saved are inverse effects on a fixed route.

Quick Revision

If the units are clean and the actual distance condition is identified correctly, speed-time-distance becomes a structured formula chapter rather than a guessing chapter.