A postal employee drives a delivery truck over the route shown in Fig. E1.25. Use the method of components to determine the magnitude and direction of her resultant displacement. In a vector-addition diagram (roughly to scale), show that the resultant displacement found from your diagram is in qualitative agreement with the result you obtained by using the method of components.
Solution 34E Consider this as the path travelled by the postal employee. Since she is moving towards the right top corner, we can infer her direction is towards north-east. Step 1: According to the law of vector addition, we can consider, AB + BC = AC 2 2 So, the magnitude of vector AC will be, A| = |B +BC + 2AB×BC cos Magnitude of AB = 2.6 km Magnitude of BC = 4 km Angle between the vectors, = 90° 2 2 2 Therefore, |AC | (2.6 km) + (4km) = 6.76 + 16 = 22.6 km = 4.77 km We know that, AC + CD = AD Vector AD is the resultant of vectors what we need and the magnitude will give us the total displacement of the postal employee from A to D. Provided, magnitude of CD = 3.1 km in the diagram Therefore, |D | A + CD + 2AC × CD cos = DCA = DCB + ACB DCB = 180° - DCF DCF = 45° DCB = 180° - 45° = 135° From the diagram, tan ( ACB) = 2.6 km / 4 km = 0.65 ACB = tan (0.65) = 33° = DCA = DCB + ACB = 135° + 33° = 168° Putting this back in the equation for magnitude. 2 2 |AD | 22.76 km + (3.1km) + 2 × 4.77 × 3.1 × cos (180 168) Cos 12 = 0.9781 We should take the angle value as (180 - ) if it is more than 90°. |AD | 22.76 km + 9.61km + (2 × 4.77 × 3.1 × 0.9781) = 32.37 + 28.93 = 61. km 2 So, the displacement for the postal employee from starting point till ending point will be 7.83 km.