91Ó°ÊÓ

To understand the motion of an object in space, we often start by looking at its position vector, which describes its placement in terms of coordinates in three-dimensional space. However, the velocity vector gives us more dynamic information about how this position changes over time. Simply put, the velocity vector is the derivative of the position vector with respect to time. This means it tells us how fast and in what direction the object is moving at any given moment.

For example, consider a position vector in the form of \( \mathbf{r}(t) = (1 + 3t)\mathbf{i} + (t - 2)\mathbf{j} - 3t\mathbf{k} \). To find the velocity vector \( \mathbf{v}(t) \), we take its derivative, resulting in \( \mathbf{v}(t) = 3\mathbf{i} + \mathbf{j} - 3\mathbf{k} \). This tells us that the object moves 3 units along the x-axis, 1 unit along the y-axis, and -3 units along the z-axis for each unit of time. Understanding this vector is crucial for determining subsequent motion characteristics like acceleration.

Just as velocity is the rate of change of the position vector, acceleration is the rate of change of the velocity vector. It tells us how the velocity itself changes with time, indicating any increase, decrease, or direction change in speed.

To calculate the acceleration vector \( \mathbf{a}(t) \), we differentiate the velocity vector. In our example, given \( \mathbf{v}(t) = 3 \mathbf{i} + \mathbf{j} - 3 \mathbf{k} \), its derivative is \( \mathbf{a}(t) = 0 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} \). This result shows us the acceleration is 0, meaning the velocity is constant. In physics, a constant velocity implies straight-line motion with no changes in speed, which simplifies analyzing the object's path and forces involved.

When analyzing motion, it is sometimes helpful to break the acceleration vector into components that align with the path of motion: the tangent and normal components. These components provide insights into the nature of the motion.

The tangent component \( a_{\mathrm{T}} \) is aligned with the velocity vector and indicates changes in speed along the path. It is calculated by differentiating the magnitude of the velocity. In cases of constant velocity like ours, \( a_{\mathrm{T}} = 0 \), implying no change in speed.

On the other hand, the normal component \( a_{\mathrm{N}} \) is perpendicular to the velocity and represents curvature or change in direction along the path. It can be found using the formula \( a_{\mathrm{N}} = \sqrt{||\mathbf{a}(t)||^2 - a_{\mathrm{T}}^2} \). Since \( \mathbf{a}(t) = 0 \) and \( a_{\mathrm{T}} = 0 \) in our example, \( a_{\mathrm{N}} \) also results in 0.

This analysis informs us that the motion is not only straight but also uniform, with the vector components reflecting the lack of any directional changes.