To solve this problem, we need to determine the speed at which the train must travel the second 100 km to achieve an average speed of 70 km/h for the entire journey.

Step 1: Understanding the problem

• Total distance between Delhi and Agra: 200 km.
• First half distance (from Delhi to the halfway point): 100 km.
• Speed for the first half: 50 km/h.
• Second half distance (from the halfway point to Agra): 100 km.
• Desired average speed for the entire journey: 70 km/h.

Step 2: Calculate the time taken for the entire journey at the average speed

The total time $T_{\text{total}}$ needed to cover 200 km at an average speed of 70 km/h is given by: $T_{\text{total}} = \frac{\text{Total distance}}{\text{Average speed}} = \frac{200 \text{ km}}{70 \text{ km/h}} = \frac{200}{70} \text{ hours} \approx 2.857 \text{ hours}$

Step 3: Calculate the time taken for the first half of the journey

The time $T_1$ taken to cover the first 100 km at 50 km/h is: $T_1 = \frac{100 \text{ km}}{50 \text{ km/h}} = 2 \text{ hours}$

Step 4: Calculate the remaining time for the second half

To maintain an average speed of 70 km/h, the total journey time should be approximately 2.857 hours. The remaining time $T_2$ for the second half of the journey is: $T_2 = T_{\text{total}} - T_1 = 2.857 \text{ hours} - 2 \text{ hours} = 0.857 \text{ hours} \approx 51.43 \text{ minutes}$

Step 5: Determine the required speed for the second half of the journey

The required speed $v_2$ for the second 100 km is given by: $v_2 = \frac{\text{Distance for the second half}}{\text{Time for the second half}} = \frac{100 \text{ km}}{0.857 \text{ hours}} \approx 116.7 \text{ km/h}$

Conclusion

To average 70 km/h for the entire journey, the train must travel the second 100 km at approximately 116.7 km/h.

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Related Questions:

1. How would the required speed change if the first half was traveled at 60 km/h?
2. What would be the average speed if the train maintained 50 km/h throughout the journey?
3. How much time would it take to complete the entire journey if the train traveled the second half at 100 km/h?
4. What if the train travels the first 100 km at 40 km/h? What speed is needed for the second half?
5. If the total distance were 300 km instead of 200 km, what speed would be required to maintain a 70 km/h average?

Tip:

Always remember that average speed is not simply