91Ó°ÊÓ

When we think about motion, the first thing that comes to mind is how we can describe where an object is at any given time. This is where position vectors come into play. A position vector is a mathematical tool that gives us the exact location of a particle in space relative to a given point, often described using coordinates. In the exercise, the position of a particle moving along the parabola described by the curve equation \( y = x^2 + 1 \) is captured by the vector function \( \mathbf{r}(t) = t \mathbf{i} + (t^2 + 1) \mathbf{j} \). This tells us that at any time \( t \), the particle's position can be found by plugging \( t \) into the equation to get an ordered pair in the plane.
This ordered pair is essentially broken down into the x-component, \( t \mathbf{i} \), and the y-component, \( (t^2 + 1) \mathbf{j} \). Each component represents movement along the x or y axis.

• The x-component \( t \mathbf{i} \) indicates motion along the horizontal x-axis.
• The y-component \( (t^2 + 1) \mathbf{j} \) reflects the particle's position along the vertical y-axis, which is also influenced by \( t \).

Understanding how position vectors work provides a foundation for analyzing motion in any dimension.

The concept of a velocity vector builds on the notion of a position vector. While a position vector tells us where a particle is, a velocity vector reveals how the particle's position changes over time. It's the derivative of the position vector with respect to time. For our exercise, we find that the velocity vector is \( \mathbf{v}(t) = 1 \mathbf{i} + 2t \mathbf{j} \).
This new vector indicates not just movement but **speed** and **direction**. The i-component (\(1 \mathbf{i}\)) remains constant, showing a steady motion in the x-direction, while the j-component changes with time \( 2t \mathbf{j} \), reflecting acceleration along the y-axis.

• A constant \( 1 \mathbf{i} \) means the particle moves consistently at one unit per time along x.
• The y-component \( 2t \mathbf{j} \) suggests that vertical speed increases as time passes.

To really grasp velocity, remember it is always tangent to the path of motion, offering insights into where the particle will be shortly.

Where velocity vectors explain speed and direction, acceleration vectors reveal how velocity itself changes over time. In our exercise, the acceleration vector is calculated as \( \mathbf{a}(t) = 0 \mathbf{i} + 2 \mathbf{j} \). This tells us that acceleration is no longer a function of time in the x-direction, but remains constant in the y-direction.
Key insights into acceleration include:

• \( 0 \mathbf{i} \) indicates there is no change in velocity in the x-direction; so, the motion here is uniform and unchanging horizontally.
• \( 2 \mathbf{j} \) means there is a constant acceleration in the y-direction, influencing how velocity increases vertically.

Acceleration helps us understand the forces driving changes in velocity. So, if you're noticing a curve moving faster upward, that's likely due to this constant vertical acceleration at work. In essence, acceleration vectors provide the final piece in mapping out a particle's motion through the entire motion sequence.