Question

You are flying on a commercial airline on your way from Houston, Texas, to Oklahoma City, Oklahoma. Your pilot announces that the plane is directly over Austin, Texas, traveling at a constant speed of \(245 \mathrm{mph}\), and will be flying directly over Dallas, Texas, \(362 \mathrm{~km}\) away. How long will it be before you are directly over Dallas, Texas?

Short answer

Answer: __________ hours (or __________ minutes)

Step-by-step solution

Convert distance from kilometers to miles

To make the units consistent, let's convert the distance from kilometers to miles. We can use the conversion factor: 1 mile = 1.60934 kilometers. So, let's convert 362 km to miles.$$ \text{Distance in miles} = \frac{362}{1.60934} $$

Calculate the time taken to travel the distance

Now that we have distance in miles, we can use the time-distance-speed formula:$$\text{Time} = \frac{\text{Distance}}{\text{Speed}}$$Plug in the values we have:\(\text{Time} = \frac{\text{Distance in miles}}{245 \text{ mph}}\)

Simplify the expression and find the time

Simplify the expression for time:$$\text{Time} = \frac{\frac{362}{1.60934}}{245}$$Calculate the time:$$\text{Time} = \frac{362}{1.60934\times245}$$After calculating the time, we get the result in hours.

Convert the time to minutes (optional)

It might be useful to convert the time to minutes. To do this, multiply the time in hours by 60:$$\text{Time}(\text{in minutes}) = \text{Time}(\text{in hours}) \times 60$$

Key concepts

Unit Conversion

Unit conversion is a crucial skill in physics and everyday life. In this exercise, we need to convert distance from kilometers to miles to match the speed units provided, which are in miles per hour (mph).

To do this, we use the conversion factor where 1 mile = 1.60934 kilometers. This is key to switching between metric and imperial systems. Converting helps us solve problems accurately by making sure all units are consistent.

• Start by dividing the distance in kilometers by the conversion factor: \( \frac{362 \text{ km}}{1.60934} \)
• This converts 362 km into a distance in miles, which is more compatible with our mph speed unit.

Remembering such conversion factors, or quickly accessing them, is very helpful. Accurate conversion ensures that our subsequent calculations will be correct.

Time-Distance-Speed Relation

The relationship between time, distance, and speed is fundamental in physics, helping us understand how objects move. The formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) simplifies this relationship.

Here's how we use it in the exercise:

• First, ensure all units are consistent. In this case, convert kilometers to miles so both distance and speed are in miles.
• Next, apply the formula: by dividing the converted distance by the speed, you calculate the time taken to cover that distance at the given speed.

This approach provides a clear method to determine how long a journey will take based on constant speed, which can be used in various real-life situations, from estimating travel times to planning logistics.

Mathematical Calculations

Mathematical calculations are the backbone of solving physics problems efficiently. Once the units are consistent, calculating the time requires straightforward arithmetic using our formula.

To find the time for the journey:

• Divide the converted distance in miles by the constant speed of 245 mph.
• Simplify the expression: \( \text{Time} = \frac{\frac{362}{1.60934}}{245} \)
• This results in a time in hours, making it easy to understand the duration needed.

Optionally, converting the time from hours to minutes could provide a more intuitive understanding. Multiply the time in hours by 60 to achieve this.
This step-by-step calculation process is a clear demonstration of how proper mathematical operations lead to solving physics questions correctly and successfully.