Problem 102 - Introductory Chemistry Atoms First

Question

Suppose it takes you \(0.850\) weeks to reach the moon in a space ship. How many seconds does it take for you to get there? Use unit analysis to calculate your answer, and show your work.

Step-by-Step Solution

Answer

It takes 514080 seconds to reach the moon in a space ship.

Step 1: Write down the given time duration in weeks

The given time duration is \(0.850\) weeks.

Step 2: Conversion factors

To convert the time duration from weeks to seconds, we need the following conversion factors: 1. There are 7 days in 1 week. 2. There are 24 hours in 1 day. 3. There are 60 minutes in 1 hour. 4. There are 60 seconds in 1 minute.

Step 3: Convert from weeks to days

Multiply the given time duration by the first conversion factor (7 days per week): \(0.850 \, \text{weeks} \times \frac{7 \, \text{days}}{1 \, \text{week}} = 5.95 \, \text{days} \)

Step 4: Convert from days to hours

Next, multiply the result obtained in step 3 by the second conversion factor (24 hours per day): \(5.95 \, \text{days} \times \frac{24 \, \text{hours}}{1 \, \text{day}} = 142.8 \, \text{hours} \)

Step 5: Convert from hours to minutes

Multiply the result obtained in step 4 by the third conversion factor (60 minutes per hour): \(142.8 \, \text{hours} \times \frac{60 \, \text{minutes}}{1 \, \text{hour}} = 8568 \, \text{minutes} \)

Step 6: Convert from minutes to seconds

Finally, multiply the result obtained in step 5 by the fourth conversion factor (60 seconds per minute): \(8568 \, \text{minutes} \times \frac{60 \, \text{seconds}}{1 \, \text{minute}} = 514080 \, \text{seconds} \)

Step 7: Write the final answer

It takes 514080 seconds to reach the moon in a space ship.

Key Concepts

Conversion Factors

When faced with unit conversion problems, an understanding of conversion factors is crucial. A conversion factor is a numerical value or equation used to multiply or divide a quantity to express it in different units without changing its actual value. It's like changing the clothes of a number - the number itself doesn't change, but the way it's presented does.

For example, in the exercise where a trip to the moon takes 0.850 weeks, we use several conversion factors to convert weeks into seconds. These factors are a sequence of equivalent measures that progressively break down the original measurement (weeks) into the target unit (seconds). Each step is a simple multiplication that employs a conversion factor such as 7 days per week, 24 hours per day, 60 minutes per hour, and 60 seconds per minute. It's important to set these factors up so that units cancel out appropriately, ensuring the final unit is what you need it to be, just as we cancel out the old clothes to dress the number in a new outfit.

Time Unit Conversion

Time unit conversion often seems daunting, but it's a matter of using the correct tools - namely conversion factors - and understanding the relationship between the units. With time, we're fortunate to have standardized conversion factors that are universally recognized.

The process involves a series of multiplications or divisions where each step uses a specific time unit conversion factor. In the original exercise, to modify the journey time from weeks to seconds, we have conversion factors at every level of the time hierarchy - from weeks to days, days to hours, hours to minutes, and finally, minutes to seconds. Remembering the sequence and the number of each smaller unit in the larger unit (like 60 seconds in 1 minute) is essential for accurate conversion. Understanding this hierarchy and its consistent pattern is key in making time conversions second nature.

Dimensional Analysis

Dimensional analysis, also known as unit analysis, is the powerhouse of solving unit conversion problems. It's a systematic way of checking and ensuring that your final answer is expressed in the correct unit of measure. It’s fundamental to accurately conducting conversions and maintaining consistency across scientific calculations.

In dimensional analysis, units are treated as algebraic quantities that can cancel each other out when they are the same. You arrange conversion factors so that the unit you want to discard is in the denominator and the unit you want to keep is in the numerator. This way, when you multiply your original measure by the conversion factors, you can clearly see how units cancel out step by step. This method is clearly demonstrated in the exercise, where we convert the given time in weeks to seconds, by methodically using conversion factors and canceling out units until only the desired unit - seconds - remains.