91Ó°ÊÓ

In the study of projectile motion, the concept of "Initial Velocity" is fundamental. When an object begins its journey, the speed and direction at which it is launched are critical initial conditions. For our example, the rabbit starts running on the ice with an initial velocity of \(6.00 \ \mathrm{m/s}\) due east. This means that from the very beginning, the rabbit is moving at a constant speed in the x direction. We express this initial motion in unit-vector notation as \(6.00\ \hat{i} \ \mathrm{m/s}\).
This vector notation helps us understand the direction—east here is represented by the \(\hat{i}\) component of the vector. By clearly identifying the initial velocity, we set the stage for calculating subsequent changes in speed and direction, especially when other forces, such as wind, come into play. Understanding initial velocity as a vector quantity allows for a clearer appreciation of how multiple influences interact over time.

"Constant Acceleration" is a key player in altering an object's velocity over time. Acceleration occurs when a force is applied, causing the object to speed up, slow down, or change direction. In the case of the rabbit, a wind force provides an acceleration of \(1.40 \ \mathrm{m/s^2}\) northward, characterized in unit-vector form as \(1.40\ \hat{j}\ \text{m/s}^2\).
Understanding acceleration in vector form is crucial, as it signifies both magnitude and direction—north in this scenario is represented by \(\hat{j}\). Because this acceleration is constant, it simplifies our calculations, allowing us to use the formula \( \vec{v}(t) = \vec{v}_0 + \vec{a} t \) to find the velocity at any given time. This means that after every instant, the rabbit's velocity in the northward direction increases by a constant amount every second—1.40 meters per second.
In projectile motion, the consistency of acceleration often relates to gravity, but in this example, it's the consistent wind force causing deviation from the initial straight path.

The "Position Vector" describes the object's location at a given time. For the rabbit, which started at the origin of the coordinate system, the position at any minute can be calculated using the formula: \[ \vec{r}(t) = \vec{r}_0 + \vec{v}_0 t + \frac{1}{2} \vec{a} t^2 \]The initial position \(\vec{r}_0 \) is zero, since the rabbit begins sliding at the origin. The term \(\vec{v}_0 t\) accounts for the distance covered due to initial velocity, while \(\frac{1}{2} \vec{a} t^2\) represents the additional distance due to acceleration.
For example, after 3 seconds, these calculations show the rabbit is at position \(18.00 \hat{i} + 6.30 \hat{j} \ \mathrm{m}\). This vector indicates movement 18 meters east and 6.3 meters north due to the initial velocity and influence of constant acceleration. Understanding the position vector provides a comprehensive picture of motion, incorporating the influences of speed, time, and directional forces.