91Ó°ÊÓ

In the study of particle motion, the velocity vector is a fundamental concept. It's derived from the position vector by taking the derivative with respect to time. This provides specific insights into the particle's direction and speed at any given moment.
The velocity vector, denoted by \( \mathbf{v}(t) \), is crucial for understanding how a particle moves. It comprises two components: one for the x-direction and one for the y-direction.
By differentiating the components of the position vector \( \mathbf{r}(t) = \frac{t}{t+1} \mathbf{i} + \frac{1}{t} \mathbf{j} \), we find each component of the velocity vector:

• For the x-component: \( \frac{d}{dt}\left(\frac{t}{t+1}\right) = \frac{1}{(t+1)^2} \)
• For the y-component: \( \frac{d}{dt}\left(\frac{1}{t}\right) = -\frac{1}{t^2} \)

This yields the velocity vector \( \mathbf{v}(t) = \frac{1}{(t+1)^2} \mathbf{i} - \frac{1}{t^2} \mathbf{j} \). By inserting a specific time, such as \( t = -\frac{1}{2} \), we easily compute the velocity at that moment, offering an accurate snapshot of the particle's speed and direction.

Acceleration provides insight into how the particle's velocity changes over time. The acceleration vector is obtained by further differentiating the velocity vector with respect to time. This step is critical in understanding the changes in motion.
Let's examine the calculation:
First, take the derivative of each component of the velocity vector \( \mathbf{v}(t) = \frac{1}{(t+1)^2} \mathbf{i} - \frac{1}{t^2} \mathbf{j} \). The results are:

• For the x-component: \( \frac{d}{dt}\left(\frac{1}{(t+1)^2}\right) = -\frac{2}{(t+1)^3} \)
• For the y-component: \( \frac{d}{dt}\left(-\frac{1}{t^2}\right) = \frac{2}{t^3} \)

The acceleration vector \( \mathbf{a}(t) = -\frac{2}{(t+1)^3} \mathbf{i} + \frac{2}{t^3} \mathbf{j} \) reflects how fast the velocity is increasing or decreasing along each axis.
At \( t = -\frac{1}{2} \), this vector helps illustrate whether the particle is speeding up or slowing down, giving critical context to its current motion dynamics.

The trajectory of a particle outlines the path it follows in the plane, described by an equation that relates \(x\) and \(y\) coordinates. This path is often represented graphically.
For the given particle position \( \mathbf{r}(t) = \frac{t}{t+1} \mathbf{i} + \frac{1}{t} \mathbf{j} \), the x and y positions are expressed individually:

• \( x = \frac{t}{t+1} \)
• \( y = \frac{1}{t} \)

To determine the trajectory: - Solve for \( t \) in terms of \( x \), from the x equation: \( t = \frac{x}{1-x} \).- Substitute \( t \) in the y equation: \( y = \frac{1-x}{x} \).Finally, this gives the trajectory equation as \( y \cdot x = 1 - x \), summarizing the particle's path and facilitating visualization in a coordinate system. Recognizing this equation empowers students to predict and plot the particle's journey across time.