Calculating Average Speed for a Two-Part Journey

To find the average speed for the entire journey, we need to calculate the total distance covered and the total time taken. The journey is described in two parts, each with a different speed and duration or distance.

Understanding the Journey Segments

The journey consists of two distinct parts:

• Part 1: Shyam drives 30 km at a speed of 45 km/h.
• Part 2: Shyam drives for 1 hour 20 minutes at a speed of 51 km/h.

Calculations for Part 1

In the first part of the journey, we are given the distance and the speed. We need to find the time taken for this part.

• Distance ($D_1$) = 30 km
• Speed ($S_1$) = 45 km/h
• Time ($T_1$) = Distance / Speed

Using the formula, the time taken for Part 1 is:

\( T_1 = \frac{D_1}{S_1} = \frac{30 \text{ km}}{45 \text{ km/h}} \)

\( T_1 = \frac{30}{45} \text{ hours} = \frac{2}{3} \text{ hours} \)

Calculations for Part 2

In the second part of the journey, we are given the time and the speed. We need to find the distance covered in this part.

• Time ($T_2$) = 1 hour 20 minutes
• Speed ($S_2$) = 51 km/h
• Distance ($D_2$) = Speed × Time

First, let's convert the time for Part 2 into hours. 20 minutes is equal to \( \frac{20}{60} \) hours = \( \frac{1}{3} \) hours.

So, \( T_2 = 1 \text{ hour} + \frac{1}{3} \text{ hours} = \frac{3}{3} \text{ hours} + \frac{1}{3} \text{ hours} = \frac{4}{3} \text{ hours} \)

Now, using the formula, the distance covered in Part 2 is:

\( D_2 = S_2 \times T_2 = 51 \text{ km/h} \times \frac{4}{3} \text{ hours} \)

\( D_2 = \frac{51 \times 4}{3} \text{ km} = 17 \times 4 \text{ km} = 68 \text{ km} \)

Calculating Total Distance and Total Time

To find the average speed for the entire journey, we sum the distances and the times from both parts.

• Total Distance ($D_{total}$) = Distance from Part 1 + Distance from Part 2
• Total Time ($T_{total}$) = Time from Part 1 + Time from Part 2

Total Distance:

\( D_{total} = D_1 + D_2 = 30 \text{ km} + 68 \text{ km} = 98 \text{ km} \)

Total Time:

\( T_{total} = T_1 + T_2 = \frac{2}{3} \text{ hours} + \frac{4}{3} \text{ hours} = \frac{2 + 4}{3} \text{ hours} = \frac{6}{3} \text{ hours} = 2 \text{ hours} \)

Calculating Average Speed

The formula for average speed is:

\( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \)

Using the calculated total distance and total time:

\( \text{Average Speed} = \frac{98 \text{ km}}{2 \text{ hours}} \)

\( \text{Average Speed} = 49 \text{ km/h} \)

The average speed for Shyam's entire journey is 49 km/h.

Average Speed = \( \frac{\text{Total Distance}}{\text{Total Time}} = \frac{98 \text{ km}}{2 \text{ h}} = 49 \text{ km/h} \)

Revision Table: Key Concepts in Average Speed

Additional Information on Average Speed Problems

When tackling problems involving average speed, especially those with multiple segments like Shyam's journey, always remember the fundamental principle: calculate the total distance and divide by the total time. Do not simply average the given speeds, as this is only correct if the time taken for each segment is the same.

Key steps often involve:

• Identifying the different segments of the journey.
• Determining (or calculating) the distance and time for each segment.
• Ensuring all units (distance and time) are consistent (e.g., all in km and hours, or all in meters and seconds). Convert units if necessary (like minutes to hours).
• Summing up the distances of all segments to get the total distance.
• Summing up the times of all segments to get the total time.
• Applying the average speed formula using the total distance and total time.

These steps help ensure you correctly calculate the average speed, accounting for variations in speed and duration across different parts of the journey.