Using only dimensional analysis, convert 65 miles per hour to kilometers per second.
The Correct Answer and Explanation is :
To convert 65 miles per hour (mph) to kilometers per second (km/s) using dimensional analysis, we can follow a systematic approach. We will break the conversion down into manageable parts by converting miles to kilometers and hours to seconds.
Step 1: Convert miles to kilometers
We start with the conversion factor for miles to kilometers. The commonly accepted conversion is:
[
1 \text{ mile} = 1.60934 \text{ kilometers}
]
Step 2: Convert hours to seconds
Next, we need to convert hours to seconds. There are 60 minutes in an hour and 60 seconds in a minute, so:
[
1 \text{ hour} = 60 \text{ minutes} \times 60 \text{ seconds/minute} = 3600 \text{ seconds}
]
Step 3: Set up the conversion
Now we can set up our dimensional analysis:
[
65 \text{ mph} = 65 \text{ miles/hour} \times \frac{1.60934 \text{ kilometers}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}}
]
Step 4: Perform the calculation
We will calculate the result step by step:
- Convert miles to kilometers:
[
65 \text{ miles/hour} \times 1.60934 \text{ kilometers/mile} = 104.6071 \text{ kilometers/hour}
] - Convert hours to seconds:
[
104.6071 \text{ kilometers/hour} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = \frac{104.6071}{3600} \text{ kilometers/second} \approx 0.0290 \text{ kilometers/second}
]
Conclusion
Thus, ( 65 \text{ miles/hour} ) is approximately ( 0.0290 \text{ kilometers/second} ).
This method of dimensional analysis relies on using known conversion factors to ensure that the units properly cancel out, leading to the desired unit of kilometers per second. This approach not only provides the correct numerical conversion but also reinforces the importance of unit consistency in physics and engineering calculations. By systematically converting each unit, we ensure the integrity of the final result.