Time and distance

📘 Time, Speed & Distance

🔹 Basic Concept

The basic relation between speed, distance, and time is:

\[ \text{Distance (D)} = \text{Speed (S)} \times \text{Time (T)} \]

\[ \text{Speed (S)} = \frac{\text{Distance (D)}}{\text{Time (T)}} \]

\[ \text{Time (T)} = \frac{\text{Distance (D)}}{\text{Speed (S)}} \]

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🔹 Average Speed

\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]

\[ \text{If equal distances: } \text{Average Speed} = \frac{2 \times S_1 \times S_2}{S_1 + S_2} \]

\[ \text{If equal times: } \text{Average Speed} = \frac{S_1 + S_2}{2} \]

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🔹 Relative Speed

Same direction:
\[ V_{\text{rel}} = |V_1 - V_2| \] \[ \text{Time to meet} = \frac{\text{Distance between them}}{V_{\text{rel}}} \]

Opposite direction:
\[ V_{\text{rel}} = V_1 + V_2 \] \[ \text{Time to meet} = \frac{\text{Distance between them}}{V_{\text{rel}}} \]

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🧠 Exam Shortcuts & Tricks

Shortcut 1: Convert km/h → m/s: multiply by \(\frac{5}{18}\), m/s → km/h: multiply by \(\frac{18}{5}\)

Shortcut 2: Time = Distance ÷ Speed

Shortcut 3: Relative speed = sum or difference depending on direction

Shortcut 4: Average speed shortcut formulas for equal distance and equal time

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✍️ Solved Examples

Example 1:
A car travels 120 km at 60 km/h. Find time taken. \[ T = \frac{D}{S} = \frac{120}{60} = 2 \text{ hours} \]

Example 2 (Average Speed, equal distances):
A train travels 100 km at 50 km/h and next 100 km at 25 km/h. Find average speed. \[ \text{Average Speed} = \frac{2 \times 50 \times 25}{50 + 25} = \frac{2500}{75} \approx 33.33 \text{ km/h} \]

Example 3 (Relative Speed, opposite direction):
Two trains move towards each other at 70 km/h and 30 km/h. Distance = 200 km. Time to meet: \[ V_{\text{rel}} = 70 + 30 = 100 \text{ km/h} \] \[ T = \frac{200}{100} = 2 \text{ hours} \]

Example 4 (Relative Speed, same direction):
Two trains move in the same direction at 60 km/h and 40 km/h. Distance between them = 120 km. Time to meet: \[ V_{\text{rel}} = 60 - 40 = 20 \text{ km/h} \] \[ T = \frac{120}{20} = 6 \text{ hours} \]

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