Question

A car travels at 22.0 mph for 15.0 min and 35.0 mph for \(30.0 \mathrm{~min}\). How far does it travel overall? a) \(23.0 \mathrm{~m}\) b) \(3.70 \cdot 10^{4} \mathrm{~m}\) c) \(1.38 \cdot 10^{3} \mathrm{~m}\) d) \(3.30 \cdot 10^{2} \mathrm{~m}\)

Short answer

(Use the conversion 1 mile = 1,609.34 meters.) a) \(3.50 \cdot 10^{4} \mathrm{~m}\) b) \(3.70 \cdot 10^{4} \mathrm{~m}\) c) \(3.80 \cdot 10^{4} \mathrm{~m}\) d) \(4.00 \cdot 10^{4} \mathrm{~m}\) Answer: b) \(3.70 \cdot 10^{4} \mathrm{~m}\)

Step-by-step solution

Convert time from minutes to hours

To be consistent with the speed given in mph (miles per hour), we need to convert the given time for both segments from minutes to hours. To do this, we simply divide the time by 60, since there are 60 minutes in an hour. - For the first segment: 15.0 minutes / 60 = 0.25 hours - For the second segment: 30.0 minutes / 60 = 0.50 hours

Calculate the distance for each segment

We will use the formula distance = speed x time to find the distance traveled in both segments. - For the first segment: distance = 22.0 mph x 0.25 hours = 5.50 miles - For the second segment: distance = 35.0 mph x 0.50 hours = 17.50 miles

Find the total distance traveled

Now, we add the distances for both segments to get the total distance traveled by the car. Total distance = 5.50 miles + 17.50 miles = 23.0 miles

Convert the total distance to meters

Since the answer choices are given in meters, we need to convert our total distance from miles to meters. We know that 1 mile is equal to 1,609.34 meters. So, we can multiply our total distance in miles by this conversion factor. Total distance in meters = 23.0 miles * 1,609.34 meters/mile ≈ 37,000 meters

Find the correct answer choice

Now, we compare our answer to the given options. We found the total distance to be approximately 37,000 meters, which is closest to option b (\(3.70 \cdot 10^{4} \mathrm{~m}\)). The correct answer is b) \(3.70 \cdot 10^{4} \mathrm{~m}\).

Key concepts

Converting Units

Understanding unit conversion is essential in physics, especially when calculating distances. When dealing with time, speed, and distance, it is important to ensure that all units are consistent to avoid errors. For example, speeds may be given in miles per hour, but time might be in minutes or even seconds. To accurately calculate distances, we must convert all units to align—typically to the base International System of Units (SI).

In our exercise, the car's speeds were given in miles per hour (mph), and the time was in minutes. To make them compatible, we converted time into hours since the speed was per hour. To do this, we divided the number of minutes by 60, as there are 60 minutes in an hour. Knowing that an hour is the base unit for the given speed (mph), this conversion is vital for accurate calculations. Similarly, distances calculated in miles must be converted to meters to match the answer options, utilizing the conversion factor where 1 mile equals 1,609.34 meters.

Speed-Time-Distance Relationship

The relationship between speed, time, and distance is one of the cornerstones of kinematics in physics. The basic equation connecting these three variables is: \[ \text{Distance} = \text{Speed} \times \text{Time} \].This formula allows us to solve problems where we need to find any one of the variables, as long as the other two are known.

In our case, we used this relationship to find the distance traveled by the car in each segment of the journey. By multiplying the speed of the car by the time traveled, which we had already converted to hours, we found the distance in miles. Then, we added the distances of both segments together to find the total distance covered by the car. This step-by-step approach is essential for breaking down complex problems into more manageable parts.

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause such motion. It involves analyzing the positions, velocities, and accelerations of moving objects. In our problem, we looked at a simple case where kinematics principles apply to calculate the total distance traveled by a car moving at constant speeds over different time intervals.

By applying the fundamental kinematics equation for distance calculation, and ensuring correct unit conversions, we determined the car's total travel distance, showcasing a practical application of kinematics in a real-world scenario. Understanding these concepts together forms a strong foundation for more advanced studies in physics and mechanics, where variables may change over time and motion may be influenced by various forces.