15. Relative Speed Problems

 15.  Relative Speed Problems

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Below are 10 relative speed problems, each with a step-by-step solution procedure and detailed explanation. Relative speed is the speed of one object relative to another, calculated as:
  • Same direction: Relative speed = |Speed of object 1 - Speed of object 2|
  • Opposite direction: Relative speed = Speed of object 1 + Speed of object 2
These problems involve scenarios like trains, boats, people, or other objects moving relative to each other, covering passing, catching up, or meeting. Units are typically km/h for speeds, km for distances, and hours for time, with conversions to m/s and meters for smaller scales (e.g., train lengths). Each problem emphasizes the application of relative speed to calculate time, distance, or other variables.
Problem 1: Two Trains Passing (Opposite Directions)
Two trains, 150 m and 200 m long, approach each other at 54 km/h and 72 km/h. How long does it take for them to pass completely?
Step-by-Step Solution:
  1. Identify given values:
    • Train 1: Length = 150 m, Speed = 54 km/h
    • Train 2: Length = 200 m, Speed = 72 km/h
  2. Calculate total distance: 150 + 200 = 350 m
  3. Calculate relative speed (opposite direction): 54 + 72 = 126 km/h
  4. Convert speed to m/s: 126 × 5/18 = 35 m/s
  5. Calculate time: Time = Distance / Relative speed = 350 / 35 = 10 seconds
  6. State result: The trains take 10 seconds to pass.
Detailed Explanation: When moving in opposite directions, the relative speed is the sum of the trains’ speeds, as they approach each other faster. The total distance is the sum of their lengths, as both trains must clear each other. Converting speed to m/s aligns units with the distance (meters), and dividing gives the time from when their fronts meet to when their rears pass.
Answer: 10 seconds
Problem 2: Two Trains Passing (Same Direction)
Two trains, 120 m and 180 m long, travel in the same direction at 60 km/h and 48 km/h. How long does it take for the faster train to pass the slower one?
Step-by-Step Solution:
  1. Identify given values:
    • Faster train: Length = 120 m, Speed = 60 km/h
    • Slower train: Length = 180 m, Speed = 48 km/h
  2. Calculate total distance: 120 + 180 = 300 m
  3. Calculate relative speed (same direction): 60 - 48 = 12 km/h
  4. Convert speed to m/s: 12 × 5/18 = 10/3 m/s
  5. Calculate time: Time = 300 / (10/3) = 300 × 3/10 = 90 seconds
  6. State result: The faster train takes 90 seconds to pass.
Detailed Explanation: In the same direction, the relative speed is the difference, as the faster train gains on the slower one slowly. The total distance (sum of lengths) is covered at this relative speed. The time is longer than in opposite directions due to the smaller relative speed, reflecting the gradual overtaking process.
Answer: 90 seconds
Problem 3: Train Passing a Person
A train 200 m long travels at 72 km/h toward a person walking at 5 km/h in the opposite direction. How long does it take to pass the person?
Step-by-Step Solution:
  1. Identify given values:
    • Train: Length = 200 m, Speed = 72 km/h
    • Person: Speed = 5 km/h
  2. Calculate relative speed (opposite direction): 72 + 5 = 77 km/h
  3. Convert speed to m/s: 77 × 5/18 = 385/18 ≈ 21.39 m/s
  4. Calculate time: Time = 200 / (385/18) = 200 × 18/385 ≈ 9.35 seconds
  5. State result: The train takes approximately 9.35 seconds.
Detailed Explanation: The person’s speed adds to the train’s speed since they move toward each other. The distance is the train’s length (person’s size is negligible). Converting to m/s ensures unit consistency, and the time reflects how quickly the train passes due to the high relative speed.
Answer: ≈ 9.35 seconds
Problem 4: Train Catching Up
A train A travels at 50 km/h. After 1 hour, train B starts at 80 km/h in the same direction from the same point. How long does it take for train B to catch up with train A?
Step-by-Step Solution:
  1. Identify given values:
    • Train A: Speed = 50 km/h, Head start = 1 hour
    • Train B: Speed = 80 km/h
  2. Calculate head start distance: 50 × 1 = 50 km
  3. Calculate relative speed (same direction): 80 - 50 = 30 km/h
  4. Calculate time: Time = 50 / 30 = 5/3 ≈ 1.67 hours
  5. State result: Train B takes approximately 1.67 hours.
Detailed Explanation: Train A’s head start creates a 50 km gap. The relative speed is the difference, as train B gains on train A. The time to close the gap is the distance divided by relative speed, assuming catching up occurs when fronts align (lengths negligible for simplicity).
Answer: ≈ 1.67 hours
Problem 5: Boats Meeting on a Lake
Two boats start 60 km apart on a lake, moving toward each other at 12 km/h and 18 km/h. How long until they meet?
Step-by-Step Solution:
  1. Identify given values:
    • Distance = 60 km
    • Boat 1 speed = 12 km/h
    • Boat 2 speed = 18 km/h
  2. Calculate relative speed (opposite direction): 12 + 18 = 30 km/h
  3. Calculate time: Time = 60 / 30 = 2 hours
  4. State result: The boats meet in 2 hours.
Detailed Explanation: In still water, the boats approach each other at the sum of their speeds. The time to meet is the initial distance divided by relative speed, assuming they meet when their positions coincide (sizes negligible). This is similar to trains meeting but in a different context.
Answer: 2 hours
Problem 6: Person and Bicycle (Same Direction)
A person walks at 4 km/h, and a bicycle travels at 20 km/h in the same direction. If the bicycle is 1 km behind, how long does it take to catch up?
Step-by-Step Solution:
  1. Identify given values:
    • Person speed = 4 km/h
    • Bicycle speed = 20 km/h
    • Initial distance = 1 km
  2. Calculate relative speed (same direction): 20 - 4 = 16 km/h
  3. Calculate time: Time = 1 / 16 = 1/16 hours
  4. Convert to minutes: 1/16 × 60 = 3.75 minutes
  5. State result: The bicycle takes 3.75 minutes.
Detailed Explanation: The bicycle gains on the person at the relative speed. The initial 1 km gap is closed by dividing the distance by relative speed. Converting to minutes makes the result practical, showing the quick catch-up due to the large speed difference.
Answer: 3.75 minutes
Problem 7: Car Passing a Truck
A car 5 m long travels at 90 km/h, and a truck 15 m long travels at 60 km/h in the same direction. How long does it take for the car to pass the truck completely?
Step-by-Step Solution:
  1. Identify given values:
    • Car: Length = 5 m, Speed = 90 km/h
    • Truck: Length = 15 m, Speed = 60 km/h
  2. Calculate total distance: 5 + 15 = 20 m
  3. Calculate relative speed (same direction): 90 - 60 = 30 km/h
  4. Convert speed to m/s: 30 × 5/18 = 25/3 m/s
  5. Calculate time: Time = 20 / (25/3) = 20 × 3/25 = 12/5 = 2.4 seconds
  6. State result: The car takes 2.4 seconds.
Detailed Explanation: The car passes the truck when it covers both their lengths (20 m) at the relative speed. Converting to m/s aligns units, and the time is short due to the small distance and significant speed difference, typical for vehicle passing scenarios.
Answer: 2.4 seconds
Problem 8: Train and Platform with Relative Motion
A train 300 m long travels at 108 km/h toward a platform 200 m long, which is moving at 18 km/h in the opposite direction (on a moving walkway). How long does it take to pass the platform?
Step-by-Step Solution:
  1. Identify given values:
    • Train: Length = 300 m, Speed = 108 km/h
    • Platform: Length = 200 m, Speed = 18 km/h
  2. Calculate total distance: 300 + 200 = 500 m
  3. Calculate relative speed (opposite direction): 108 + 18 = 126 km/h
  4. Convert speed to m/s: 126 × 5/18 = 35 m/s
  5. Calculate time: Time = 500 / 35 ≈ 14.29 seconds
  6. State result: The train takes approximately 14.29 seconds.
Detailed Explanation: The platform’s motion (unusual but possible in theoretical problems) increases the relative speed, as they move toward each other. The total distance is the sum of lengths, and the time is calculated as usual, showing how relative motion applies to non-standard objects.
Answer: ≈ 14.29 seconds
Problem 9: Delayed Start Meeting
Two cars start 120 km apart, moving toward each other at 40 km/h and 50 km/h. The second car starts 10 minutes late. How long after the second car starts do they meet?
Step-by-Step Solution:
  1. Identify given values:
    • Distance = 120 km
    • Car 1 speed = 40 km/h
    • Car 2 speed = 50 km/h
    • Delay = 10 minutes = 10/60 = 1/6 hours
  2. Calculate distance covered by car 1 during delay: 40 × 1/6 = 20/3 ≈ 6.67 km
  3. Calculate remaining distance: 120 - 6.67 ≈ 113.33 km
  4. Calculate relative speed (opposite direction): 40 + 50 = 90 km/h
  5. Calculate time to meet: Time = 113.33 / 90 ≈ 1.26 hours
  6. Add delay: Total time = 1/6 + 1.26 ≈ 1.43 hours
  7. State result: They meet approximately 1.43 hours after the second car starts.
Detailed Explanation: Car 1 reduces the initial distance during the delay. The remaining distance is covered at the relative speed (sum of speeds). Total time includes the delay and meeting time, showing how delays affect relative motion problems.
Answer: ≈ 1.43 hours
Problem 10: Relative Speed with Wind
An airplane flies at 500 km/h in still air. With a 50 km/h tailwind, it travels 600 km. With a 50 km/h headwind, it travels 600 km. Find the time difference between the two trips.
Step-by-Step Solution:
  1. Identify given values:
    • Airplane speed = 500 km/h
    • Wind speed = 50 km/h
    • Distance = 600 km each way
  2. Calculate speeds:
    • With tailwind (same direction): 500 + 50 = 550 km/h
    • With headwind (opposite direction): 500 - 50 = 450 km/h
  3. Calculate times:
    • Tailwind time = 600 / 550 = 12/11 ≈ 1.0909 hours
    • Headwind time = 600 / 450 = 4/3 ≈ 1.3333 hours
  4. Calculate time difference: 4/3 - 12/11 = (48 - 36) / 33 = 12/33 = 4/11 ≈ 0.3636 hours
  5. State result: The time difference is approximately 0.36 hours (or 21.8 minutes).
Detailed Explanation: The tailwind increases the airplane’s speed (like same direction), while the headwind decreases it (like opposite direction). Each trip’s time is distance divided by effective speed. The difference shows the wind’s impact, with the headwind trip taking longer due to lower relative speed.
Answer: ≈ 0.36 hours (or 21.8 minutes)
Summary
These 10 relative speed problems cover diverse scenarios: trains passing in opposite or same directions, catching up, people, vehicles, boats, airplanes with wind, and even a moving platform. Each uses relative speed (sum for opposite directions, difference for same direction) to calculate time or distance. Step-by-step solutions ensure clarity, with unit conversions (e.g., km/h to m/s) for consistency. Detailed explanations highlight why relative speed matters, how object lengths or delays affect results, and common pitfalls (e.g., not summing lengths for passing). These problems build a strong understanding of relative motion in various contexts.

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