Calculating Average Speed A Car's Journey At Varying Velocities
Calculating average speed can sometimes be tricky, especially when an object travels different distances at different speeds. The problem presented involves a car that covers two-thirds of its journey at 60 m/s and the remaining one-third at a slower 20 m/s. To accurately determine the average speed for the entire trip, we must consider both the distances traveled at each speed and the time taken for each segment. This is not a simple average of the two speeds; instead, it requires a weighted calculation that accounts for the proportion of the journey covered at each speed. In this comprehensive discussion, we will explore the step-by-step method to solve this problem, ensuring a clear understanding of the underlying physics principles.
The key to solving this problem lies in recognizing that average speed is defined as the total distance traveled divided by the total time taken. Mathematically, this is expressed as: Average Speed = Total Distance / Total Time. To apply this formula, we need to break down the car's journey into its constituent parts. The car travels two distinct segments: the first covering 2/3 of the total distance at 60 m/s, and the second covering 1/3 of the total distance at 20 m/s. We don't know the actual total distance, but we can represent it with a variable, say 'D'. This allows us to express the distances of the two segments as (2/3)D and (1/3)D, respectively. The next step involves calculating the time taken for each segment, which can be done using the relationship: Time = Distance / Speed. This foundational concept from physics helps us link distance, speed, and time, allowing us to compute the total time for the journey and, consequently, the average speed.
In summary, to find the average speed, we will first calculate the time taken for each segment of the journey using the formula Time = Distance / Speed. For the first segment, the distance is (2/3)D and the speed is 60 m/s. For the second segment, the distance is (1/3)D and the speed is 20 m/s. Once we have the time for each segment, we can add them together to find the total time. Then, we divide the total distance, D, by the total time to obtain the average speed. This method ensures that we correctly account for the different speeds and distances, providing an accurate representation of the car's overall average speed for the entire journey. The following sections will delve into the detailed calculations, step by step, to arrive at the final answer.
Step-by-Step Solution
1. Define the Variables
To begin, let’s clearly define our variables. Let the total distance be represented by D. This allows us to express the distances of the two segments in terms of D. The first segment covers 2/3 of the total distance, which is (2/3)D, and the second segment covers 1/3 of the total distance, which is (1/3)D. Understanding these distances is crucial for the subsequent calculations.
2. Calculate Time for the First Segment
To calculate the time taken for the first segment, we use the formula: Time = Distance / Speed. In this segment, the distance is (2/3)D and the speed is 60 m/s. Plugging these values into the formula, we get:
Time1 = ((2/3) * D) / 60 = (2D) / (3 * 60) = D / 90 seconds. This equation gives us the time taken to cover the first segment in terms of the total distance D.
3. Calculate Time for the Second Segment
Similarly, we calculate the time taken for the second segment using the same formula, Time = Distance / Speed. For this segment, the distance is (1/3)D and the speed is 20 m/s. Therefore,
Time2 = ((1/3) * D) / 20 = D / 60 seconds. This tells us the time taken to cover the second segment, also in terms of the total distance D.
4. Calculate Total Time
Now that we have the time taken for each segment, we can find the total time by adding them together:
Total Time = Time1 + Time2 = (D / 90) + (D / 60)
To add these fractions, we need a common denominator, which in this case is 180. So, we rewrite the equation as:
Total Time = (2D / 180) + (3D / 180) = (5D / 180) seconds. Simplifying this, we get Total Time = D / 36 seconds. This is the total time taken for the entire journey, expressed in terms of the total distance D.
5. Calculate Average Speed
Finally, we can calculate the average speed using the formula: Average Speed = Total Distance / Total Time. The total distance is D, and the total time is D / 36 seconds. Plugging these values into the formula, we get:
Average Speed = D / (D / 36) = D * (36 / D)
Notice that the total distance D cancels out, leaving us with:
Average Speed = 36 m/s. Therefore, the average speed of the car for the entire journey is 36 m/s.
Answer and Options
The correct answer is A) 36 m/s. This result highlights the importance of considering the time spent at each speed when calculating average speed, rather than simply averaging the speeds themselves. The weighted average, which takes into account the distances traveled at each speed, provides a more accurate representation of the car's overall speed during the journey.
Common Pitfalls and How to Avoid Them
When tackling problems involving average speed, several common pitfalls can lead to incorrect answers. Understanding these pitfalls and how to avoid them is crucial for accurate problem-solving. One of the most frequent mistakes is simply averaging the speeds without considering the time spent at each speed. This approach is only valid if the object travels at each speed for the same amount of time, which is not the case in this scenario. In our problem, the car travels at 60 m/s for 2/3 of the distance and at 20 m/s for the remaining 1/3, meaning it spends different amounts of time at each speed.
To illustrate the error of simple averaging, let's consider what would happen if we incorrectly applied this method. If we were to average 60 m/s and 20 m/s, we would get (60 + 20) / 2 = 40 m/s. This is significantly different from the correct answer of 36 m/s. The reason this approach fails is that it doesn't account for the fact that the car spends more time traveling at the slower speed (20 m/s) because that segment covers a proportionally larger fraction of the total time. Another common mistake is confusing average speed with average velocity. While both involve the total distance or displacement and the total time, velocity is a vector quantity that includes direction, whereas speed is a scalar quantity that only considers magnitude.
To avoid these pitfalls, it is essential to follow a structured approach. First, always remember the fundamental definition of average speed: Total Distance / Total Time. Break down the journey into segments and calculate the time spent in each segment using the formula Time = Distance / Speed. Be meticulous in your calculations and ensure that all units are consistent. When calculating total time, correctly add the times for each segment, and when finding the average speed, make sure to divide the total distance by the total time. By following these steps and understanding the underlying concepts, you can confidently solve average speed problems and avoid common errors.
Real-World Applications of Average Speed Calculations
Understanding and calculating average speed is not just an academic exercise; it has numerous real-world applications across various fields. From transportation planning to sports analytics, the concept of average speed helps us make informed decisions and understand the dynamics of motion. In transportation, average speed calculations are essential for planning travel routes and estimating arrival times. For example, when mapping a road trip, navigation systems use average speed data for different road types (highways, city streets, etc.) to provide estimated times of arrival. This helps travelers plan their journey effectively, accounting for potential delays due to traffic or road conditions.
In the field of logistics and delivery services, calculating average speed is crucial for optimizing delivery routes and schedules. Companies need to determine the most efficient routes for their vehicles to minimize delivery times and fuel consumption. By analyzing average speeds on different routes at various times of the day, logistics managers can make data-driven decisions to improve operational efficiency and reduce costs. This involves considering factors such as traffic patterns, road construction, and delivery schedules to ensure timely and cost-effective service.
Sports is another area where average speed calculations play a significant role. In track and field events, the average speed of a runner helps assess performance and compare athletes. For instance, calculating the average speed of a 100-meter sprinter provides a clear indication of their speed and can be used for performance analysis and training optimization. Similarly, in team sports like soccer or basketball, understanding the average speed of players during a game can provide insights into their physical conditioning, stamina, and overall contribution to the team's performance. Sports analysts use this data to develop strategies and training programs aimed at enhancing athletic performance.
Furthermore, average speed calculations are used in scientific research to study the movement of animals, the flow of traffic, and various other phenomena. Researchers can track the movement patterns of animals over time and calculate their average speeds to understand migration patterns, foraging behavior, and habitat use. In traffic engineering, average speed data is used to assess road network efficiency, identify congestion hotspots, and develop traffic management strategies. By understanding how average speed is calculated and applied in these diverse contexts, we can appreciate its practical significance and its role in solving real-world problems.
