91Ó°ÊÓ

In physics, the concept of a position vector is essential when describing where an object is located in space. A position vector points from the origin of a coordinate system to the exact location of the object. It's a useful tool for illustrating an object's position relative to a defined point, often the origin. For example, if we're dealing with an ion, and its initial position vector is noted as \( \vec{r}_i = 5.0 \hat{\mathrm{i}} - 6.0 \hat{\mathrm{j}} + 2.0 \hat{\mathrm{k}} \), this tells us the ion's location in a three-dimensional space. Here,

• \(5.0 \hat{\mathrm{i}} \) represents the x-component,
• \(-6.0 \hat{\mathrm{j}} \) the y-component, and
• \(2.0 \hat{\mathrm{k}} \) the z-component.

The use of unit vectors \, \( \hat{\mathrm{i}}, \hat{\mathrm{j}}, \hat{\mathrm{k}} \,\) makes it easier to understand and visualize the direction and magnitude of the position changes in each dimension.

Understanding the change in position of an object involves recognizing how far the object has moved from its initial point to its final point. This change can be represented as a vector as well, termed the "change in position vector" or displacement vector, \( \Delta \vec{r} \). Mathematically, it's the difference between the final position vector \( \vec{r}_f \) and the initial position vector \( \vec{r}_i \).
In this exercise, the ion's final position is \( \vec{r}_f = -2.0 \hat{\mathrm{i}} + 8.0 \hat{\mathrm{j}} - 2.0 \hat{\mathrm{k}} \). To find the displacement, use the formula:

• \( \Delta \vec{r} = \vec{r}_f - \vec{r}_i \).

Carry out the subtraction component-wise:

• \( \Delta x = -2.0 - 5.0 = -7.0 \),
• \( \Delta y = 8.0 + 6.0 = 14.0 \),
• \( \Delta z = -2.0 - 2.0 = -4.0 \).

So, the change in position vector is \( \Delta \vec{r} = -7.0 \hat{\mathrm{i}} + 14.0 \hat{\mathrm{j}} - 4.0 \hat{\mathrm{k}} \). This tells you how the ion moved in each spatial direction during the time interval.

Once you understand the position vector and change in position, calculating the average velocity is straightforward. Average velocity provides information about how fast and in what direction an object is moving over a specified time period. It's determined by dividing the change in position vector by the time interval during which the change occurred.
The equation to use is:

• \( \vec{v}_{\text{avg}} = \frac{\Delta \vec{r}}{\Delta t} \).

Given that the time interval, \( \Delta t \), is 10 seconds, and the displacement \( \Delta \vec{r} \) is \( -7.0 \hat{\mathrm{i}} + 14.0 \hat{\mathrm{j}} - 4.0 \hat{\mathrm{k}} \), the average velocity is calculated as follows:

• \( \vec{v}_{\text{avg}} = \frac{-7.0 \hat{\mathrm{i}} + 14.0 \hat{\mathrm{j}} - 4.0 \hat{\mathrm{k}}}{10} \),
• which simplifies to \( -0.7 \hat{\mathrm{i}} + 1.4 \hat{\mathrm{j}} - 0.4 \hat{\mathrm{k}} \).

This vector tells us the ion's average velocity in the x, y, and z directions over the 10 seconds, showing both its speed and direction.