Time Speed and Distance: Notes और MCQ Practice

आसान भाषा में

Time, Speed, Distance का relation बहुत simple है – एक ही formula से सब solve होते हैं।

मुख्य Formulas / Rules

• Distance = Speed × Time
• Speed = Distance ÷ Time
• Time = Distance ÷ Speed
• Relative Speed (towards each other) = Speed₁ + Speed₂
• Relative Speed (same direction) = |Speed₁ − Speed₂|
• Average Speed for equal distances = (2 × S1 × S2) ÷ (S1 + S2)

Quick Tricks

1. Speed‑Time‑Distance shortcut: अगर distance known है और speed दो भागों में दिया है (जैसे 30 km/h for 2 hrs, 40 km/h for 3 hrs) तो total distance = (30×2)+(40×3). Directly add distance = speed × time for each segment.
2. Percentage change in speed: New Speed = Old Speed × (1 + %change/100). Use this to quickly find new time = Old Distance ÷ New Speed.

Common Mistakes

1. Units mix‑up – km को m में या hour को minute में convert किए बिना calculation करना.
2. Equal distance problems में average speed को (S1+S2)/2 ले लेना; सही है harmonic mean = (2×S1×S2)/(S1+S2).

Concept

Time‑Speed‑Distance problems deal with the relationship between how far an object travels (distance), how fast it moves (speed), and how long it takes (time). The basic idea is that distance = speed × time. By rearranging the formula you can find any one of the three when the other two are known.

Important formulas / rules / facts

• Distance = Speed × Time
• Speed = Distance ÷ Time
• Time = Distance ÷ Speed
• Average speed = Total distance ÷ Total time
• Relative speed (same direction) = Faster speed – Slower speed
• Relative speed (opposite direction) = Faster speed + Slower speed
• Convert units before using the formulas (e.g., 1 km = 1000 m, 1 h = 3600 s)
• When a boat/plane crosses a river/air‑current, its resultant speed is found using Pythagoras if the motion is not along the same line.

Solved Examples

Example 1

A train travels 120 km at 60 km/h and then 180 km at 45 km/h. Find its average speed for the whole journey.

Solution:

1. Find time for each part.

• t1 = 120 ÷ 60 = 2 h
• t2 = 180 ÷ 45 = 4 h

1. Total distance = 120 + 180 = 300 km
2. Total time = 2 + 4 = 6 h
3. Average speed = Total distance ÷ Total time = 300 ÷ 6 = 50 km/h

Example 2

Two cyclists start from the same point. Cyclist A is 5 km ahead of Cyclist B. Their speeds are 12 km/h and 15 km/h respectively. After how many hours will Cyclist B overtake Cyclist A?

Solution:

1. Relative speed = 15 – 12 = 3 km/h (since they move in the same direction)
2. Gap to close = 5 km
3. Time = Gap ÷ Relative speed = 5 ÷ 3 = 1.666… h = 1 h 40 min

Example 3

A boat can row at 10 km/h in still water. The river flows at 4 km/h. The river is 500 m wide. How long will it take the boat to reach the opposite bank directly across?

Solution:

1. Convert width to km: 500 m = 0.5 km
2. Speed across the river (perpendicular component) = √(10² – 4²) = √(100 – 16) = √84 ≈ 9.165 km/h
3. Time = Distance ÷ Speed = 0.5 ÷ 9.165 ≈ 0.0545 h
4. Convert to seconds: 0.0545 × 3600 ≈ 196 s (about 3 min 16 s)

Example 4

A man walks 8 km at 4 km/h and returns the same distance at 6 km/h. Find his average speed for the round trip.

Solution:

1. Time forward = 8 ÷ 4 = 2 h
2. Time return = 8 ÷ 6 = 1.333… h
3. Total distance = 16 km
4. Total time = 2 + 1.333… = 3.333… h
5. Average speed = 16 ÷ 3.333… = 4.8 km/h

Example 5

Two trains start from opposite stations 300 km apart and move towards each other. Train X runs at 70 km/h and Train Y at 50 km/h. After how many minutes will they meet?

Solution:

1. Relative speed = 70 + 50 = 120 km/h
2. Time to meet = Distance ÷ Relative speed = 300 ÷ 120 = 2.5 h
3. Convert to minutes: 2.5 × 60 = 150 minutes

Shortcuts and Tricks

• Use LCM for time problems: When two speeds are given as fractions, find a common multiple of denominators to avoid messy decimals.
• Average speed shortcut: If the distances are equal, average speed = (2 × speed1 × speed2) ÷ (speed1 + speed2).
• Relative speed trick: Always decide whether objects move in the same or opposite directions before applying the relative speed formula.
• Quick conversion: 1 km/h = 5/18 m/s. Multiply by 5 and divide by 18.
• Cross‑river shortcut: When the boat’s speed in still water is much larger than the current, approximate the perpendicular component as the boat speed itself; the error is negligible for exam time.

Common Mistakes to Avoid

1. Mixing units: Never use km with minutes or meters with km/h without conversion.
2. Averaging speeds directly: Average speed is not (speed1 + speed2)/2 unless the time spent at each speed is equal.
3. Ignoring direction in relative speed: Forgetting to add speeds when objects move towards each other leads to a larger answer.
4. Using distance = speed × time incorrectly: Remember to keep the same unit for both speed and time.
5. Forgetting to convert width of river/track to km when speed is in km/h: This causes a huge error in time calculation.