Speed, Distance & Time

Speed measures how fast an object moves. It's the rate at which distance is covered with respect to time.

$$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

Conversely,

$$\text{Distance} = \text{Speed} \times \text{Time} \quad \text{and} \quad \text{Time} = \frac{\text{Distance}}{\text{Speed}}$$

Intuitive Analogy:

Imagine running on a straight track. If you run faster (higher speed), you cover more distance in less time. If you're slower, the same distance takes more time.

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2. Key Formulas & Shortcuts

Basic Units

• Speed: km/hr or m/s

• Conversion:

  • $1 \text{ km/hr} = \frac{5}{18} \text{ m/s}$
  • $1 \text{ m/s} = \frac{18}{5} \text{ km/hr}$

Relative Speed

• Same Direction: Subtract speeds

  $$\text{Relative Speed} = |S_1 - S_2|$$

• Opposite Direction: Add speeds

  $$\text{Relative Speed} = S_1 + S_2$$

Average Speed

• Different Distances:

  $$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

• Same Distance (Two Speeds):

  $$\text{Average Speed} = \frac{2S_1S_2}{S_1 + S_2}$$

Crossing Objects

• Length of object A crossing object B (length $L_1 + L_2$) moving at relative speed $S$:

  $$\text{Time to cross} = \frac{L_1 + L_2}{S}$$

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3. Conceptual Tips & Common Mistakes

• Always check unit consistency (convert all to m/s or km/hr).
• Relative speed applies when two objects are moving toward/away from each other.
• Don’t apply the average speed shortcut unless the distance is the same.
• For crossing problems, include the length of both objects.
• Roundabout questions might hide the time difference subtly—read the question twice.

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4. Visual Explanation

Example 1: Distance-Time Graph

• A straight-line graph sloping upwards indicates constant speed.
• A steeper slope → faster speed.
• Flat segment → stationary.

(We can later embed illustrative graphs on AptiDude to make this vivid.)

Example 2: Two People Walking Toward Each Other

Visualize two people on a 100-meter track walking toward each other—one at 3 m/s, another at 2 m/s. They meet in $\frac{100}{3+2} = 20$ seconds. Their combined movement shrinks the gap faster.

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5. Solved Examples

Example 1: Basic Speed-Distance

Q: A car covers 150 km in 3 hours. What is its speed?

A:

$$\text{Speed} = \frac{150}{3} = 50 \text{ km/hr}$$

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Example 2: Average Speed – Equal Distance

Q: A man travels from A to B at 60 km/hr and returns at 40 km/hr. What is his average speed?

A:

$$\text{Avg Speed} = \frac{2 \cdot 60 \cdot 40}{60 + 40} = \frac{4800}{100} = 48 \text{ km/hr}$$

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Example 3: Train Crossing a Platform

Q: A 200-meter-long train crosses a 300-meter platform in 30 seconds. Find the speed of the train.

A:
Total distance = 200 + 300 = 500 m

$$\text{Speed} = \frac{500}{30} = 16.\overline{66} \text{ m/s} = 60 \text{ km/hr}$$