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Understanding average velocity is crucial in the analysis of particle motion. It tells us how fast the particle is moving over a specific period. To find the average velocity, we need to know two main things: the displacement of the particle and the duration of the interval.
The displacement is the "straight-line" distance between the starting and ending points of the particle's journey, while the time interval is how long the journey took. Once we have these, we simply divide the displacement by the time interval. For instance, in the given problem, we calculated the average velocity by determining the displacement as \[ \| \Delta \mathbf{r} \| \approx 0.412 \text{ m} \] and dividing it by \( t = 0.02 \text{ s} \), leading to an average velocity of approximately \( v_{\text{av}} \approx 20.6 \text{ m/s} \).

• it's important to note that average velocity includes both direction and speed
• and it can differ from the instantaneous velocity, which can vary over the course of motion.

The displacement calculation helps us figure out precisely how the position of the particle changes over time. It's different from distance, as displacement considers only the initial and final positions, not the entire path taken.
To calculate displacement in two dimensions, we use vector subtraction between the final and initial position vectors. This is important to ensure any changes in both directions (x and y) are considered. Here, the displacement of the particle was determined using:\[ \Delta \mathbf{r} = (5.1 \mathbf{i} + 0.4 \mathbf{j}) - 5 \mathbf{i} = 0.1 \mathbf{i} + 0.4 \mathbf{j} \] This means the particle moved 0.1 meters in the x-direction and 0.4 meters in the y-direction.

• Remember, the displacement gives us a vector result, which includes magnitude and direction.
• Even if a particle returns to its starting point, making the net displacement zero, it could have covered some distance—it just depends entirely on the endpoints.

Determining the angle with the axis can provide insight into the particle's motion direction. This angle, often denoted as \( \theta \), is calculated using the tangent ratio derived from the components of the displacement vector.
To find \( \theta \), apply the formula: \[ \tan \theta = \frac{text Y-component}{text X-component} \] In this case, \( \tan \theta = \frac{0.4}{0.1} = 4 \) , providing the angle \( \theta \) via the inverse tangent function as \( \theta \approx 75.96^\circ \).
Understanding this angle is important:

• It informs us about the direction of particle motion relative to the x-axis.
• For example, an angle of almost 76 degrees suggests strong directional movement in the y-axis.
• It's an excellent way to visualize a velocity vector's orientation, enhancing our understanding of the particle's motion.