Calculating Train Average Speed

The question asks for the average speed of a train that travels at different speeds over different time intervals. To find the average speed, we need to calculate the total distance covered by the train and divide it by the total time taken for the journey.

The formula for average speed is:

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

The journey is divided into three segments with different speeds and durations.

Step 1: Calculate the distance covered in each segment

We know that Distance = Speed $\times$ Time. We must ensure that units are consistent. The speeds are given in km/h, and the times are given in minutes. Let's convert the time for each segment into hours.

Segment 1:

• Speed = 90 km/h
• Time = 10 minutes
• Converting time to hours: $$ 10 \text{ minutes} = 10 \times \frac{1}{60} \text{ hours} = \frac{1}{6} \text{ hours} $$
• Distance in Segment 1: $$ \text{Distance}_1 = 90 \text{ km/h} \times \frac{1}{6} \text{ h} = \frac{90}{6} \text{ km} = 15 \text{ km} $$

Segment 2:

• Speed = 60 km/h
• Time = 25 minutes
• Converting time to hours: $$ 25 \text{ minutes} = 25 \times \frac{1}{60} \text{ hours} = \frac{25}{60} \text{ hours} = \frac{5}{12} \text{ hours} $$
• Distance in Segment 2: $$ \text{Distance}_2 = 60 \text{ km/h} \times \frac{5}{12} \text{ h} = \frac{60 \times 5}{12} \text{ km} = 5 \times 5 \text{ km} = 25 \text{ km} $$

Segment 3:

• Speed = 15 km/h
• Time = 4 minutes
• Converting time to hours: $$ 4 \text{ minutes} = 4 \times \frac{1}{60} \text{ hours} = \frac{4}{60} \text{ hours} = \frac{1}{15} \text{ hours} $$
• Distance in Segment 3: $$ \text{Distance}_3 = 15 \text{ km/h} \times \frac{1}{15} \text{ h} = \frac{15}{15} \text{ km} = 1 \text{ km} $$

Step 2: Calculate the total distance

The total distance is the sum of the distances covered in each segment.

$$ \text{Total Distance} = \text{Distance}_1 + \text{Distance}_2 + \text{Distance}_3 $$

$$ \text{Total Distance} = 15 \text{ km} + 25 \text{ km} + 1 \text{ km} = 41 \text{ km} $$

Step 3: Calculate the total time

The total time is the sum of the time taken for each segment. It's easier to sum in minutes first and then convert to hours for the final calculation.

Total time in minutes = 10 minutes + 25 minutes + 4 minutes = 39 minutes.

Converting total time to hours:

$$ \text{Total Time} = 39 \text{ minutes} = 39 \times \frac{1}{60} \text{ hours} = \frac{39}{60} \text{ hours} = \frac{13}{20} \text{ hours} $$

Step 4: Calculate the average speed

Now, use the formula for average speed with the total distance and total time.

$$ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{41 \text{ km}}{\frac{13}{20} \text{ h}} $$

$$ \text{Average Speed} = 41 \times \frac{20}{13} \text{ km/h} = \frac{820}{13} \text{ km/h} $$

Step 5: Convert the average speed to a mixed fraction

To convert $\frac{820}{13}$ to a mixed fraction, divide 820 by 13.

$$ 820 \div 13 $$

$$ 820 = 13 \times 63 + 1 $$

The quotient is 63 and the remainder is 1. So, the mixed fraction is $63 \frac{1}{13}$.

Therefore, the average speed of the train is $63 \frac{1}{13}$ km/h.

Average Speed = Total Distance / Total Time = $41 \text{ km} / (13/20) \text{ h} = 41 \times 20/13 \text{ km/h} = 820/13 \text{ km/h}$

$$ \frac{820}{13} = 63 \frac{1}{13} $$

The average speed of the train is $63 \frac{1}{13}$ km/h.

Revision Table: Average Speed Calculation

Additional Information: Speed, Distance, and Time

Speed, distance, and time are fundamental concepts in physics and mathematics, often related by the formula: $$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$ This can be rearranged to find distance ($\text{Distance} = \text{Speed} \times \text{Time}$) or time ($\text{Time} = \frac{\text{Distance}}{\text{Speed}}$).

When an object travels at different speeds over different parts of its journey, the average speed is not simply the average of the speeds. It must be calculated using the total distance and the total time.

• Constant Speed: If speed is constant, the relationship is straightforward.
• Varying Speed: When speed varies, the journey is often broken down into segments. Calculate distance for each segment and sum them up. Sum the time taken for each segment. Then, use the total distance and total time to find the average speed.
• Units: Always pay close attention to units (e.g., km/h, m/s, minutes, hours) and ensure they are consistent throughout the calculation. Convert units if necessary.