Math Problem Statement

A truck goes M to N at an average speed of 60km/h . The journey takes 30hrs. It returns from N to M on the same road with an average speed of 40km/h . What was the average speed of the truck during the whole journey in m/s ?

Solution

To calculate the average speed of the truck during the whole journey in meters per second (m/s), we follow these steps:

Step 1: Total Distance Calculation

The truck's journey consists of two parts:

  1. From MM to NN: Distance from M to N=Speed×Time=60km/h×30hours=1800km.\text{Distance from } M \text{ to } N = \text{Speed} \times \text{Time} = 60 \, \text{km/h} \times 30 \, \text{hours} = 1800 \, \text{km}.

  2. From NN to MM: Since the truck returns on the same road, the distance is the same: 1800km1800 \, \text{km}.

The total distance for the whole journey: Total Distance=1800km+1800km=3600km.\text{Total Distance} = 1800 \, \text{km} + 1800 \, \text{km} = 3600 \, \text{km}.

Step 2: Total Time Calculation

  1. Time taken from MM to NN: Time from M to N=30hours.\text{Time from } M \text{ to } N = 30 \, \text{hours}.

  2. Time taken from NN to MM: Time from N to M=DistanceSpeed=1800km40km/h=45hours.\text{Time from } N \text{ to } M = \frac{\text{Distance}}{\text{Speed}} = \frac{1800 \, \text{km}}{40 \, \text{km/h}} = 45 \, \text{hours}.

The total time for the whole journey: Total Time=30hours+45hours=75hours.\text{Total Time} = 30 \, \text{hours} + 45 \, \text{hours} = 75 \, \text{hours}.

Step 3: Average Speed Calculation

The average speed of the whole journey is calculated using the formula: Average Speed=Total DistanceTotal Time.\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}.

In kilometers per hour: Average Speed=3600km75hours=48km/h.\text{Average Speed} = \frac{3600 \, \text{km}}{75 \, \text{hours}} = 48 \, \text{km/h}.

Convert this to meters per second (m/s\text{m/s}): Average Speed in m/s=48km/h×1000m1km×1hour3600seconds=480003600=13.33m/s.\text{Average Speed in } \text{m/s} = 48 \, \text{km/h} \times \frac{1000 \, \text{m}}{1 \, \text{km}} \times \frac{1 \, \text{hour}}{3600 \, \text{seconds}} = \frac{48000}{3600} = 13.33 \, \text{m/s}.

Final Answer:

The average speed of the truck during the whole journey is: 13.33m/s\boxed{13.33 \, \text{m/s}}


Would you like further clarification or more details? Here are related questions to deepen your understanding:

  1. What is the formula for converting km/h to m/s?
  2. How does the time taken affect the average speed in cases of varying speeds?
  3. What would the average speed be if the truck's return speed was 60 km/h instead of 40 km/h?
  4. Can you explain why the average speed is not the arithmetic mean of 60 km/h and 40 km/h?
  5. How would the calculation change if the truck traveled different distances for the two legs of the journey?

Tip: Always remember that the average speed for a round trip is determined by the total distance divided by the total time, not the arithmetic mean of the speeds.

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Math Problem Analysis

Mathematical Concepts

Average Speed
Unit Conversion
Distance-Time-Speed Relationship

Formulas

Total Distance = Distance (M to N) + Distance (N to M)
Total Time = Time (M to N) + Time (N to M)
Average Speed = Total Distance / Total Time
Conversion from km/h to m/s: 1 km/h = 5/18 m/s

Theorems

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Suitable Grade Level

Grades 8-10