91Ó°ÊÓ

Velocity vectors are graphical representations of an object's speed and direction. They are usually depicted with arrows, where the length of the arrow represents the magnitude (speed), and the arrowhead points in the direction of motion. Velocity vectors are essential in understanding how an object is moving in a plane.
In the original exercise, we have two critical times: \(t_1\) and \(t_2\). At \(t_1\), the velocity vector is expressed with components \(v_{x1} = 90 \, \mathrm{m/s}\) and \(v_{y1} = 110 \, \mathrm{m/s}\), situating this vector in the first quadrant, where both x and y components are positive. At \(t_2\), the vector components change to \(v_{x2} = -170 \, \mathrm{m/s}\) and \(v_{y2} = 40 \, \mathrm{m/s}\), placing this vector in the second quadrant, as the x component is negative while the y remains positive.
This change in direction and magnitude can significantly affect the net motion of the jet, showcasing different orientations at two points in time.

Acceleration components are crucial in understanding how an object's velocity changes over time in each direction. These components are evaluated separately for each axis, which makes it easier to analyze changes in motion.
To find the average acceleration components, we take the difference between the velocity components at \(t_2\) and \(t_1\), and then divide by the time interval \(t_2 - t_1\).

• The x-component of acceleration \(a_{x_{avg}}\) is calculated as: \[ a_{x_{avg}} = \frac{v_{x2} - v_{x1}}{t_2 - t_1} = -8.67 \, \mathrm{m/s^2} \] which shows a significant decrease, indicating slowed motion in the x direction.
• The y-component \(a_{y_{avg}}\) follows similarly: \[ a_{y_{avg}} = \frac{v_{y2} - v_{y1}}{t_2 - t_1} = -2.33 \, \mathrm{m/s^2} \]

These values suggest changes in how much the jet speeds up or slows down along each axis.

Magnitude of acceleration gives us a scalar measure of how quickly the velocity changes, combining the effects in both directions into a single quantity.
Using the Pythagorean theorem, we calculate the magnitude of the average acceleration vector as follows:
\[ a_{avg} = \sqrt{a_{x_{avg}}^2 + a_{y_{avg}}^2} = \sqrt{(-8.67 \, \mathrm{m/s^2})^2 + (-2.33 \, \mathrm{m/s^2})^2} \approx 8.97 \, \mathrm{m/s^2} \] This calculation indicates that the jet experiences a substantial average acceleration magnitude over the given time interval.
The result gives you a sense of how much the overall speed of the jet is changing, without regard to the specific direction of the change.

Direction of acceleration is necessary to understand the precise orientation of the forces acting on the object. It gives us insight into how direction changes when speed changes.
We determine the direction using the tangent of the angle \(\theta\) that the acceleration vector makes with the negative x-axis, calculated as:
\[ \theta = \tan^{-1}\left(\frac{a_{y_{avg}}}{a_{x_{avg}}}\right) \approx 15.1^\circ \] Since both acceleration components are negative, this angle must be adjusted for the third quadrant:

• The direction of the average acceleration is \(180^\circ + 15.1^\circ = 195.1^\circ\) relative to the positive x-axis.

Understanding this angle helps us visualize how the path of the jet is influenced, showing a trend of turning towards the third quadrant, reflecting both the direction and decline in horizontal motion.