Understanding and Calculating Average Velocity in Vector Mechanics
When dealing with particle motion and vector mechanics, understanding how to calculate average velocity is fundamental. In this article, we will walk through a detailed example to provide clarity on the calculations involved.
The Problem: Position Vector Change of a Particle
A particle is observed to change its position from 10 meters east to 10 meters north in 1 second. We need to determine the particle's average velocity during this motion. Let's break this down step by step.
Initial and Final Positions
To begin, we need to establish the initial and final positions of the particle in Cartesian coordinates.
Initial Position: (vec{r}_i 10 ) meters east, which is equivalent to (10 -10 0) in Cartesian coordinates. Final Position: (vec{r}_f 10 ) meters north, which is equivalent to (0 10).Displacement Calculation
Displacement is the difference between the final and initial positions. It is represented by the vector (vec{d}).
Mathematically, [vec{d} vec{r}_f - vec{r}_i (0, 10) - (10, 0) (-10, 10)]
Magnitude of Displacement
The magnitude of the displacement vector is calculated using the Pythagorean theorem.
Mathematically, [|vec{d}| sqrt{(-10)^2 10^2} sqrt{100 100} sqrt{200} 10sqrt{2} meters)
Time Taken
The time taken for this displacement is given as 1 second.
Average Velocity
Average velocity is defined as the displacement divided by the time taken. Using the values we have calculated,
Mathematically, [vec{v}_{avg} frac{vec{d}}{t} frac{(-10, 10)}{1} (-10, 10) m/s)
Magnitude of Average Velocity
The magnitude of the average velocity is also calculated using the Pythagorean theorem.
Mathematically, [|vec{v}_{avg}| sqrt{(-10)^2 10^2} sqrt{100 100} sqrt{200} 10sqrt{2} m/s)
Conclusion
In summary, the average velocity of the particle is ((-10, 10)) m/s with a magnitude of (10sqrt{2} m/s).
Interpretation
The negative sign in the x-component of the average velocity indicates a westward component, while the positive y-component indicates a northward component. Therefore, the particle's average velocity can be described as moving 10√2 m/s in a direction that is north of west.
Alternative Perspectives
Another perspective suggests that the particle's average velocity is simply the change in position, which is 10 m/s north. This interpretation is valid in cases where the path taken is linear, but in this case, it does not account for the displacement's components distinctly.