The concept of average velocity can sometimes be tricky, especially when dealing with changes in direction. Let’s analyze a scenario where A Truck Traveled 400 Meters North and then 300 meters east to understand the correct approach to calculate average velocity.
Imagine a truck embarking on a journey. Initially, it moves 400 meters north over a span of 80 seconds. Subsequently, the truck changes direction and travels 300 meters east in 70 seconds. The question is: what is the magnitude of the average velocity of the truck throughout this entire trip?
The provided solution and a teacher’s alternative calculation offer differing approaches to this problem. Let’s break down each method and see why they lead to different answers.
Two Approaches to Average Velocity
The initial attempt calculates the average velocity by dividing the total distance traveled by the total time:
- Total distance (Δx) = 500 m (calculated using the Pythagorean theorem: √(400² + 300²))
- Total time (Δt) = 150 s
- Average velocity (vav) = 500/150 = 3.3 m/s
The teacher’s approach, however, involves vector components:
- vav = (300/70)i + (400/80)j = (30/7)i + 5j
- ||vav|| = 6.6 m/s
Why the Discrepancy?
The key to understanding the difference lies in the distinction between speed and velocity. The first calculation determines the average speed, while the teacher’s method calculates the magnitude of the average velocity.
Speed is a scalar quantity that refers to “how fast an object is moving.” It’s the total distance traveled divided by the total time. Velocity, on the other hand, is a vector quantity that refers to “the rate at which an object changes its position.” It considers the displacement (change in position) and direction.
In this scenario, the truck’s displacement is the straight-line distance from its starting point to its ending point, which is 500 meters. However, its average velocity considers both the northward and eastward components of its movement. The teacher’s method correctly calculates the average velocity by finding the vector sum of the velocities in each direction.
A Calculus-Based Approach
To further illustrate the concept, let’s explore a calculus-based approach. We can define the truck’s position as a function of time.
During the first 80 seconds (a truck traveled 400 meters north):
- R80(t) = (400/80)t
- V80(t) = 400/80 = 5 m/s (northward)
During the next 70 seconds (traveling east):
- R70(t) = (√(400² + 300²) – 400)/70 * t
- V70(t) = (√(400² + 300²) – 400)/70 = 1.43 m/s (eastward, relative to the northward displacement)
The average velocity can then be calculated by integrating the velocity function over time and dividing by the total time:
Vavg = (∫080 5 dy + ∫070 1.43 dx) / (70+80) = 3.33 m/s
This calculation aligns with the initial method, which focuses on total distance over total time. However, this method doesn’t account for the vector nature of velocity.
Conclusion
While the initial calculation provides the average speed, the teacher’s method offers a more accurate representation of the average velocity by considering the directional components of the truck’s movement. The magnitude of the average velocity is 6.6 m/s. When analyzing motion, it’s crucial to differentiate between speed and velocity to accurately describe an object’s movement.
