91Ó°ÊÓ

First, we need to determine the velocity vector for each segment of the trip using given speed and direction. Segment west, from Orion to Chester: Speed: \(90\,\mathrm{km/h}\) Direction: west (\(0^\circ\)) Velocity vector: \(\vec{v}_1 = 90\,\mathrm{km/h} * \cos(0) \, \mathrm{\hat{i}} + 90\,\mathrm{km/h} * \sin(0)\, \mathrm{\hat{j}}\) Segment south, from Chester to Seiling: Speed: \(80\,\mathrm{km/h}\) Direction: south (\(270^\circ\)) Velocity vector: \(\vec{v}_2 = 80\,\mathrm{km/h} * \cos(270) \, \mathrm{\hat{i}} + 80\,\mathrm{km/h} * \sin(270) \, \mathrm{\hat{j}}\) Segment southeast, from Seiling to Oakwood: Speed: \(100\,\mathrm{km/h}\) Direction: southeast (\(135^\circ\)) Velocity vector: \(\vec{v}_3 = 100\,\mathrm{km/h} * \cos(135) \, \mathrm{\hat{i}} + 100\,\mathrm{km/h} * \sin(135) \, \mathrm{\hat{j}}\)

The initial velocity vector is the velocity vector of the first segment: \(\vec{v}_i = \vec{v}_1\) The final velocity vector is the velocity vector of the last segment: \(\vec{v}_f = \vec{v}_3\)

The change in velocity can be calculated as the difference between the final and initial velocity vectors: \(\Delta \vec{v} = \vec{v}_f - \vec{v}_i\)

We need to determine the time spent during each segment of the trip to calculate the total time. Time spent in segment 1 (west): \(t_1 = \frac{16 \mathrm{km}}{90\,\mathrm{km/h}}\) Time spent in segment 2 (south): \(t_2 = \frac{8 \mathrm{km}}{80\,\mathrm{km/h}}\) Time spent in segment 3 (southeast): \(t_3 = \frac{34 \mathrm{km}}{100\,\mathrm{km/h}}\)

The total time can be calculated by adding the time for each segment: \(t_{total} = t_1 + t_2 + t_3\) Finally, we can calculate the average acceleration using the change in velocity and the total time: \(\vec{a}_{avg} = \frac{\Delta \vec{v}}{t_{total}}\) Now you can plug in the numbers and compute the change in velocity and average acceleration for the given problem.