Problem 16 - Physics for Scientists and Engineers with Modern Physics

Question

An airplane travels at \(950 \mathrm{~km} / \mathrm{h}\). How long does it take to travel \(1.00 \mathrm{~km} ?\)

Step-by-Step Solution

Answer

It takes approximately 0.00105 hours or about 3.78 seconds.

Step 1: Understand the Problem

We need to find out how long it takes the airplane to travel 1 kilometer, given that its speed is 950 km/h.

Step 2: Write Down the Speed Formula

The formula to find time when you know the speed and distance is: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \). Here, distance is 1 km and speed is 950 km/h.

Step 3: Substitute the Values

Substitute the values in the formula: \( \text{Time} = \frac{1 \text{ km}}{950 \text{ km/h}} \).

Step 4: Calculate the Time

Calculate the quotient: \( \text{Time} = \frac{1}{950} \) hours. This equals approximately 0.00105 hours.

Step 5: Convert Hours to Seconds (optional)

If needed, convert 0.00105 hours into seconds by multiplying by 3600 (since 1 hour = 3600 seconds): \( 0.00105 \times 3600 \approx 3.78 \) seconds.

Key Concepts

Speed Formula

When determining how long it takes to travel a certain distance, the speed formula is essential. This formula helps us relate distance traveled to time and speed. The speed formula is given by:
\[ \text{Speed} = \frac{\text{Distance}}{\text{Time}} \]
If we know two of these three variables, we can always find the third. In the given exercise, we want to find time, so we rearrange the formula to:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]
This rearrangement allows us to solve for time by dividing the distance by speed, which is exactly how the problem is approached in the solution.Before using this formula, it's important to ensure that the speed and distance are in compatible units, which leads us directly to the next key concept.

Unit Conversion

Often in physics, converting units is a crucial step to ensure that mathematical calculations are accurate. In this exercise, the speed of the airplane is given in kilometers per hour (km/h), and the distance is in kilometers (km). Luckily, these units are already compatible since both use kilometers for length.However, when required, time results might need to be understood in different units, like seconds. Unit conversion involves multiplying the amount you have by a conversion factor. For example, 1 hour equals 3600 seconds. To convert time from hours to seconds:
- Multiply the number of hours by 3600.
For instance, 0.00105 hours can be converted to seconds as follows:
\[ 0.00105 \times 3600 = 3.78 \text{ seconds} \]
This makes understanding results in practical, everyday units easier.

Distance-Time Relationship

In kinematics, the distance-time relationship is a fundamental concept, demonstrating how the position of an object changes over time when it moves at a constant speed. The formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) describes this linear relationship between distance traveled and the time taken, given a constant speed.In practical terms, when the speed is high, the time required to travel a given distance is short. Conversely, if your speed is low, you will need more time to cover the same distance. In our example:

• The airplane travels at 950 km/h, which means it covers a long distance in a short period.
• To travel just 1 km, the time taken is a tiny fraction of an hour, specifically about 0.00105 hours.

Understanding the distance-time relationship helps in analyzing not just how fast something is moving, but also how long it will take to reach a destination at that speed.