Problem 73 - Chemistry

Question

Among many alternative units that might be considered as a measure of time is the shake rather than the second. Based on the expression "faster than a shake of a lamb's tail" we'll define 1 shake as equal to \(2.5 \times 10^{-4} \mathrm{~s}\). If a car is traveling at \(55 \mathrm{mi} / \mathrm{h}\), what is its speed in \(\mathrm{cm} /\) shake?

Step-by-Step Solution

Answer

The car's speed is approximately 0.614 cm/shake.

Step 1: Understand the Given Units

We are provided the speed of the car in miles per hour (\( 55 \, \mathrm{mi/h} \) ) and we need to convert it to centimeters per shake. A shake is given as\( 2.5 \times 10^{-4} \, \mathrm{s} \).

Step 2: Convert Speed to Centimeters per Second

First, convert \( 55 \, \mathrm{mi/h} \) to centimeters per second. 1 mile equals 160934.4 centimeters and 1 hour equals 3600 seconds. Therefore:\[ 55 \, \mathrm{mi/h} = 55 \, \mathrm{mi/h} \times \frac{160934.4 \, \mathrm{cm}}{1 \, \mathrm{mi}} \times \frac{1 \, \mathrm{h}}{3600 \, \mathrm{s}} = \frac{55 \times 160934.4}{3600} \, \mathrm{cm/s} \]

Step 3: Calculate Speed in Centimeters per Shake

Using the conversion factor for shakes (\( 1 \, \text{shake} = 2.5 \times 10^{-4} \, \mathrm{s} \)), convert the speed from centimeters per second to centimeters per shake:\[ \text{Speed in cm/shake} = \left( \text{Speed in cm/s} \right) \times \left( 2.5 \times 10^{-4} \right) \] Insert the speed in cm/s calculated in the previous step to find the speed in cm/shake.

Step 4: Perform the Calculation

First, calculate \( 55 \, \mathrm{mi/h} \) to cm/s:\[ 55 \, \mathrm{mi/h} = \frac{55 \times 160934.4}{3600} \approx 2454.704 \mathrm{cm/s} \]Then convert to cm/shake:\[ \text{Speed in cm/shake} = 2454.704 \times 2.5 \times 10^{-4} \approx 0.613676 \mathrm{cm/shake} \]

Step 5: Finalize the Conversion

The final calculated speed of the car in cm/shake is approximately 0.614 cm/shake, after rounding to three decimal places for simplicity.

Key Concepts

Speed Calculation

Calculating speed involves finding out how fast an object is moving over a certain distance in a given amount of time. In this problem, we are converting a car's speed: initially given in miles per hour (mi/h), into centimeters per shake. Understanding speed calculation is critical for various real-world applications, from transportation to physics experiments. To calculate speed, you divide the distance by time.

This problem requires us to convert the given units rather than just perform a straightforward division. As such, tackling it involves using dimensional analysis and conversion factors to switch between different units. By breaking it down into two main conversions, you simplify the process: first from miles per hour to centimeters per second, and then into the desired centimeters per shake.

Dimensional Analysis

Dimensional analysis is a powerful mathematical technique used to convert one unit of measurement to another. It ensures the correctness of a conversion process by maintaining the same dimensions throughout.

This process involves identifying the units for each measurement and using conversion factors to move between them while keeping track of cancellation. In the given problem, dimensional analysis helps us convert the car's speed from miles per hour to centimeters per second.

For instance, converting miles to centimeters and hours to seconds by using:

• 1 mile = 160934.4 cm
• 1 hour = 3600 seconds

Following this, we can calculate the speed in cm/s by canceling out units and ensuring the logic of the conversion consistently follows these given factors.

Conversion Factors

Conversion factors are fractions or ratios that express the same value in different units. They are central to the process of converting measurements like speed into different units.

When multiplying a measurement by a conversion factor, you change the units without changing the original value, since a conversion factor essentially equals one. In this exercise, proper use of conversion factors is key for translating speed measurements into different units.

To convert miles per hour to centimeters per second, and eventually to centimeters per shake, we rely on:

• The conversion of miles to centimeters: \( 1 \text{ mile} = 160934.4 \text{ cm} \).


• The conversion of hours to seconds: \( 1 \text{ hour} = 3600 \text{ seconds} \).


• The conversion between seconds and shakes: \( 1 \text{ shake} = 2.5 \times 10^{-4} \text{ s} \).

By strategically applying these conversion factors, the original car speed in miles per hour becomes speed in cm/shake, effectively solving the problem.