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Velocity vectors represent the speed and direction of an object in motion. In our exercise, a jet plane's velocity vectors at two different timestamps are analyzed. A velocity vector consists of two components: the x-component (horizontal) and the y-component (vertical). At time \(t_1 = 0\), the velocity vector \(v_1\) has components \((90 \text{ m/s}, 110 \text{ m/s})\), indicating motion towards the first quadrant. At \(t_2 = 30.0 \text{ s}\), the velocity vector \(v_2\) changes to \((-170 \text{ m/s}, 40 \text{ m/s})\), pointing towards the second quadrant.
The direction of a vector is illustrated by the angle it makes with the positive x-axis, and its magnitude, or size, corresponds to the length of the vector arrow. To understand how these vectors change over time, consider both the change in magnitude and direction from \(v_1\) to \(v_2\).
Understanding velocity vectors helps predict the path and speed of moving objects, which is crucial in fields like aviation and navigation.

The components of velocity break down a vector into its horizontal and vertical parts. These components are vital for calculating changes in motion over time. For the jet, at \(t_1\), the components are \(v_x = 90 \text{ m/s}\) and \(v_y = 110 \text{ m/s}\). At \(t_2\), these shift to \(v_x = -170 \text{ m/s}\) and \(v_y = 40 \text{ m/s}\).
Splitting a vector into components allows us to analyze each dimension separately, making calculations more manageable. This technique helps in applying mathematical operations like vector addition or subtraction by dealing with each component individually. This way, one can derive the total effect of multiple forces acting on an object in two-dimensional space.
Analyzing the components can thus help us determine the rate of change along each axis independently, which is particularly important in physics problems involving motion.

Acceleration measures how quickly velocity changes over time and involves both magnitude and direction. To compute the magnitude, use the Pythagorean theorem: \[|\vec{a}| = \sqrt{a_x^2 + a_y^2}\]For our jet plane, the x-component of average acceleration is computed as \(a_x = \frac{-170 \text{ m/s} - 90 \text{ m/s}}{30.0 \text{ s}} = -8.67 \text{ m/s}^2\).Similarly, for the y-component: \(a_y = \frac{40 \text{ m/s} - 110 \text{ m/s}}{30.0 \text{ s}} = -2.33 \text{ m/s}^2\).
Using these, the magnitude of the average acceleration is \(9.07 \text{ m/s}^2\).
The direction is found using the tangent function: \( an \theta = \frac{a_y}{a_x}\).Hence, \(\theta\) is calculated as \(15.1^\circ\). Because the acceleration lies in the third quadrant, adjust by adding 180° to find it is \(195.1^\circ\) from the positive x-axis. This indicates the overall direction in which the velocity vector is changing.
Understanding both the magnitude and direction of acceleration is essential for predicting future motion paths for objects, aiding better navigation and control.