To determine the average speed for the trips between San Antonio and Houston, we need to break down the journey into two parts: the trip to Houston and the return trip to San Antonio. Each part of the journey has different conditions affecting the average speed, so let’s analyze them step by step.
Calculating Average Speed from San Antonio to Houston
For the trip from San Antonio to Houston, we know that half of the time is spent traveling at 55 km/h and the other half at 90 km/h. Let’s denote the total time taken for the trip as T. Therefore, the time spent at each speed is T/2.
Distance Calculation
The distance covered during each segment can be calculated using the formula:
For the first half of the trip:
- Distance at 55 km/h = 55 × (T/2) = 27.5T
For the second half of the trip:
- Distance at 90 km/h = 90 × (T/2) = 45T
The total distance from San Antonio to Houston is the sum of both distances:
- Total Distance = 27.5T + 45T = 72.5T
Average Speed Calculation
The average speed is defined as the total distance divided by the total time:
- Average Speed = Total Distance / Total Time
Substituting the values we calculated:
- Average Speed = 72.5T / T = 72.5 km/h
Calculating Average Speed from Houston to San Antonio
On the return trip, the conditions change. Here, half of the distance is traveled at 55 km/h and the other half at 90 km/h. Let’s denote the total distance from Houston to San Antonio as D.
Distance Segmentation
We can split the distance into two equal parts:
- Distance at 55 km/h = D/2
- Distance at 90 km/h = D/2
Time Calculation
Next, we calculate the time taken for each segment:
- Time at 55 km/h = (D/2) / 55 = D/110
- Time at 90 km/h = (D/2) / 90 = D/180
The total time for the return trip is the sum of both times:
- Total Time = D/110 + D/180
To add these fractions, we need a common denominator, which is 990:
- Total Time = (9D/990) + (5.5D/990) = (14.5D/990) = D/68.96
Average Speed Calculation
Now we can find the average speed for the return trip:
- Average Speed = Total Distance / Total Time
Substituting the values:
- Average Speed = D / (D/68.96) = 68.96 km/h
Summary of Results
In summary, the average speeds for the trips are:
- From San Antonio to Houston: 72.5 km/h
- From Houston to San Antonio: 68.96 km/h
This analysis illustrates how different factors, such as time spent at varying speeds and distance traveled, can significantly affect average speed calculations. Understanding these principles can help in various real-world applications, such as planning travel times or optimizing routes.
