John drives \(16 \mathrm{km}\) directly west from Orion to Chester at a speed of \(90 \mathrm{km} / \mathrm{h},\) then directly south for \(8.0 \mathrm{km}\) to Seiling at a speed of \(80 \mathrm{km} / \mathrm{h}\), then finally $34 \mathrm{km}\( southeast to Oakwood at a speed of \)100 \mathrm{km} / \mathrm{h}$. Assume he travels at constant velocity during each of the three segments. (a) What was the change in velocity during this trip? [Hint: Do not assume he starts from rest and stops at the end.] (b) What was the average acceleration during this trip?

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.