To solve this problem, we need to determine the speed required for the second leg of the journey in order to achieve an average speed of 70 km/h for the entire 200 km trip. Let's break it down step-by-step.
### Step 1: Understand the Given Data
- Total distance of the journey: 200 km
- Distance for the first leg: 100 km
- Speed for the first leg: 50 km/h
- Desired average speed for the whole journey: 70 km/h
### Step 2: Calculate the Time Taken for the First Leg of the Journey
The time taken for the first leg can be calculated using the formula:
[tex]\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \][/tex]
For the first 100 km at 50 km/h:
[tex]\[ \text{Time for first leg} = \frac{100 \text{ km}}{50 \text{ km/h}} = 2 \text{ hours} \][/tex]
### Step 3: Calculate the Total Time Required for the Entire Journey to Achieve an Average Speed of 70 km/h
The total time required can be calculated using the formula for average speed:
[tex]\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \][/tex]
Rearranging to solve for total time:
[tex]\[ \text{Total Time} = \frac{\text{Total Distance}}{\text{Average Speed}} \][/tex]
For a total distance of 200 km and an average speed of 70 km/h:
[tex]\[ \text{Total Time Required} = \frac{200 \text{ km}}{70 \text{ km/h}} \approx 2.857 \text{ hours} \][/tex]
### Step 4: Calculate the Time Remaining for the Second Leg of the Journey
The time remaining for the second leg is the difference between the total time required and the time already spent on the first leg:
[tex]\[ \text{Time for second leg} = \text{Total Time Required} - \text{Time for first leg} \][/tex]
[tex]\[ \text{Time for second leg} \approx 2.857 \text{ hours} - 2 \text{ hours} \approx 0.857 \text{ hours} \][/tex]
### Step 5: Calculate the Speed Required for the Second Leg of the Journey
Finally, the speed required for the second leg can be calculated using the formula:
[tex]\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \][/tex]
For the second 100 km and the calculated time of 0.857 hours:
[tex]\[ \text{Speed for second leg} = \frac{100 \text{ km}}{0.857 \text{ hours}} \approx 116.667 \text{ km/h} \][/tex]
### Result
To average 70 km/h for the entire 200 km journey, the train must travel the second 100 km at a speed of approximately 116.667 km/h.
