Average Speed Calculation for Train Journey

Problem Details

• Distance for the first part ($d_1$): $100\text{ km}$
• Speed during the first part ($v_1$): $50\text{ km/h}$
• Distance for the second part ($d_2$): $100\text{ km}$
• Speed during the second part ($v_2$): $25\text{ km/h}$
• Objective: Find the average speed ($v_{avg}$) for the total journey.

Solution Steps

1. Calculate the time taken ($t_1$) for the first $100\text{ km}$.

   Using the formula: Time = Distance / Speed

   $t_1 = \frac{d_1}{v_1} = \frac{100\text{ km}}{50\text{ km/h}} = 2\text{ hours}$

2. Calculate the time taken ($t_2$) for the next $100\text{ km}$.

   Using the formula: Time = Distance / Speed

   $t_2 = \frac{d_2}{v_2} = \frac{100\text{ km}}{25\text{ km/h}} = 4\text{ hours}$

3. Determine the total distance and total time.

   Total Distance ($d_{total}$): $d_{total} = d_1 + d_2 = 100\text{ km} + 100\text{ km} = 200\text{ km}$

   Total Time ($t_{total}$): $t_{total} = t_1 + t_2 = 2\text{ hours} + 4\text{ hours} = 6\text{ hours}$

4. Calculate the average speed ($v_{avg}$) for the total journey.

   Formula: Average Speed = Total Distance / Total Time

   $v_{avg} = \frac{d_{total}}{t_{total}} = \frac{200\text{ km}}{6\text{ hours}}$

   $v_{avg} = \frac{100}{3}\text{ km/h}$

   $v_{avg} \approx 33.33\text{ km/h}$

Result

The average speed for the total journey is approximately $33.33\text{ km/h}$.