Understanding Average Velocity: A Case Study of a Boy's Journey
Have you ever wondered how to solve a real-world problem using physics principles? In this article, we'll explore a specific scenario where a boy travels from his home to a center at different speeds and returns home. This problem can help us understand the concept of average velocity, as well as the difference between speed and velocity.
What Is Average Velocity?
Before diving into the numerical solution, it's essential to understand the concept of average velocity. Average velocity is a vector quantity that describes the straight-line displacement of an object relative to time. It is the total displacement divided by the total time taken to cover that displacement. Unlike speed, which is a scalar quantity (only magnitude), velocity considers both direction and magnitude.
Problem Statement
A boy travels from his home to a center at a speed of 40 m/s and returns to his home at a speed of 60 m/s. We need to find his average velocity.
Solving the Problem
To solve this problem, we need to calculate the total distance traveled and the total time taken, and then use the formula for average velocity:
Average Velocity Total Distance / Total Time
Step 1: Define the Variables
Let's denote the distance from home to the center as d meters.
Step 2: Calculate the Time Taken for Each Leg of the Journey
The time taken to travel from home to the center at 40 m/s:
Time 1 d / 40 seconds
The time taken to travel from the center back to home at 60 m/s:
Time 2 d / 60 seconds
Step 3: Calculate the Total Time
The total time taken for the round trip:
Total Time Time 1 Time 2 (d / 40) (d / 60)
To add these fractions, find a common denominator:
Total Time (3d 2d) / 120 5d / 120 seconds
Step 4: Calculate the Total Distance
The total distance traveled is twice the distance to the center:
Total Distance 2d meters
Step 5: Calculate the Average Velocity
Substitute the total distance and total time into the formula for average velocity:
Average Velocity Total Distance / Total Time 2d / (5d / 120) (2d * 120) / (5d) 240 / 5 48 m/s
Therefore, the boy's average velocity is 48 m/s.
Another Approach and the Importance of Definitions
Some may argue that the average velocity would be zero because the boy returns to his starting point. However, this is incorrect because velocity is a vector quantity. Here, we must consider displacement, which is zero in this case. Velocity, however, is not zero since the magnitude of the displacement vector is non-zero during the journey.
Using an arbitrary distance like the least common multiple (LCM) between 60 and 40, which is 120 meters, can simplify calculations:
Calculating time steps:
Time 1 120 / 40 3 seconds Time 2 120 / 60 2 seconds Total Time 3 2 5 seconds
Total Distance 2 * 120 240 meters
Average Velocity 240 / 5 48 m/s
As we can see, this approach yields the same result as the method using variables.
Conclusion
Average velocity is a crucial concept in physics, and understanding it can help solve real-world problems. Remember, speed and velocity are different concepts. Use the correct definitions and calculations to arrive at accurate results.
By practicing such problems, you can enhance your understanding of physics and improve your problem-solving skills. Always remember to break down complex problems into simpler steps for clarity and accuracy.