Problem 47 - Prealgebra

Question

Speed of a Bullet. The bullet from a rifle leaves the barrel traveling \(1,500\) feet/second. Convert \(1,500\) feet/second to miles/hour. (Round to the nearest whole number.)

Step-by-Step Solution

Answer

The bullet travels at approximately 1,023 miles per hour.

Step 1: Understand Units of Conversion

We need to convert from feet per second (ft/s) to miles per hour (mph). First, note that there are 5,280 feet in a mile and 3,600 seconds in an hour.

Step 2: Convert Feet to Miles

To convert feet to miles, we use the conversion factor: 1 mile = 5,280 feet. So, we divide the speed in feet per second by 5,280 to get miles per second: \[ \text{Miles per second} = \frac{1,500 \, \text{ft/s}}{5,280} \approx 0.2841 \, \text{miles/s} \]

Step 3: Convert Seconds to Hours

Next, we convert the speed from miles per second to miles per hour by multiplying by the number of seconds in an hour (3,600 seconds): \[ \text{Miles per hour} = 0.2841 \, \text{miles/s} \times 3,600 \approx 1,022.76 \, \text{mph} \]

Step 4: Round to the Nearest Whole Number

Finally, round 1,022.76 miles per hour to the nearest whole number. The answer is 1,023 miles per hour.

Key Concepts

Feet to Miles Conversion

Unit conversion is a fundamental skill in understanding how to navigate between different measurement systems. When converting feet to miles, the key factor to utilize is that there are 5,280 feet in one mile. This relationship is used to convert a speed given in feet per second to a speed in miles per second. To accomplish the conversion, simply divide the number of feet by 5,280. For example, if you're working with a speed of 1,500 feet per second, it would translate to:

• Miles per second = \( \frac{1,500 \, \text{ft/s}}{5,280} \) \( \approx 0.2841 \, \text{mi/s} \)

This process is akin to breaking a large number of small units into a smaller number of larger units, which is the essence of unit conversions. Remember to perform this division as an intermediate step whenever converting feet-based rates to miles.

Seconds to Hours Conversion

Converting seconds to hours is another critical step for expressing speeds in a more conventional format, like miles per hour. There are 3,600 seconds in an hour. Thus, when converting a speed from a per-second rate to a per-hour rate, this multiplicative factor is essential. Once you've got your miles per second from the previous conversion, you should multiply by 3,600 to switch to miles per hour. For instance:

• Miles per hour = \( 0.2841 \, \text{mi/s} \times 3,600 \) \( \approx 1,022.76 \, \text{mph} \)

This step scales up the value to account for passing from smaller units of time (seconds) to larger ones (hours). Understanding that this factor shows how many times a second fits into an hour is crucial for correctly making this conversion.

Speed Calculation

Calculating speed involves translating a given rate between differing units, using conversions like feet to miles and seconds to hours. This ensures the speed is expressed in terms you can readily interpret and apply, like miles per hour, commonly used for vehicles and projectiles. Start with the original speed in feet per second. Convert feet to miles using a division, then switch from seconds to hours with a multiplication. This two-step process looks as follows:

• First, \( \frac{1,500 \, \text{ft/s}}{5,280} \approx 0.2841 \, \text{mi/s} \)
• Then, \( 0.2841 \, \text{mi/s} \times 3,600 \approx 1,022.76 \, \text{mph} \)

Even after making these conversions, rounding and interpreting results is an important part of real-world applications. Here, you would round 1,022.76 mph to 1,023 mph for a clean, practical answer. This is how speed calculations convert raw data into actionable insights.